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Double-Angle, Reduction, and Half-Angle Identities

Author: Sophia
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what's covered
In this lesson, you will use double-angle, reduction, and half-angle identities to evaluate exact values, simplify expressions, and verify trigonometric identities. Specifically, this lesson will cover:

Table of Contents

1. Using Double-Angle Identities

Using the sum of angles identities, we can establish identities that give values of cos open parentheses 2 x close parentheses comma sin open parentheses 2 x close parentheses comma and tan open parentheses 2 x close parentheses in terms of trigonometric functions of x.

Starting with the cosine, write cos open parentheses 2 x close parentheses equals cos open parentheses x plus x close parentheses.

cos open parentheses x plus x close parentheses equals cos   x   cos   x minus sin   x   sin   x Apply the sum of angles identity for the cosine of a sum of angles.
equals cos squared x minus sin squared x Simplify.

Thus, cos open parentheses 2 x close parentheses equals cos squared x minus sin squared x.

Notice that the identity contains cos squared x and sin squared x comma which are related by the Pythagorean identity sin squared x plus cos squared x equals 1.

  • Solving for cos squared x comma we have cos squared x equals 1 minus sin squared x.
  • Solving for sin squared x comma we have sin squared x equals 1 minus cos squared x.
By making some substitutions into the identity cos open parentheses 2 x close parentheses equals cos squared x minus sin squared x comma we are able to get alternate formulas for cos open parentheses 2 x close parentheses.

First, if we replace cos squared x with 1 minus sin squared x comma we have:

table attributes columnalign left end attributes row cell cos open parentheses 2 x close parentheses equals cos squared x minus sin squared x end cell row cell space space space space space space space space space space space space space equals open parentheses 1 minus sin squared x close parentheses minus sin squared x end cell row cell space space space space space space space space space space space space space equals 1 minus 2 sin squared x end cell end table

This identity gives the value of cos open parentheses 2 x close parentheses in terms of sin   x.

Similarly, if we replace sin squared x with 1 minus cos squared x comma we have yet another version of cos open parentheses 2 x close parentheses colon

table attributes columnalign left end attributes row cell cos open parentheses 2 x close parentheses equals cos squared x minus sin squared x end cell row cell space space space space space space space space space space space space space equals cos squared x minus open parentheses 1 minus cos squared x close parentheses end cell row cell space space space space space space space space space space space space space equals cos squared x minus 1 plus cos squared x end cell row cell space space space space space space space space space space space space space equals 2 cos squared x minus 1 end cell end table

This identity gives the value of cos open parentheses 2 x close parentheses when cos   x is known.

formula to know
Cosine of a Double Angle
table attributes columnalign left end attributes row cell cos open parentheses 2 x close parentheses equals cos squared x minus sin squared x end cell row cell cos open parentheses 2 x close parentheses equals 1 minus 2 sin squared x end cell row cell cos open parentheses 2 x close parentheses equals 2 cos squared x minus 1 end cell end table

To establish a double-angle identity for sine, write sin open parentheses 2 x close parentheses equals sin open parentheses x plus x close parentheses.

sin open parentheses x plus x close parentheses equals sin   x   cos   x plus cos   x   sin   x Apply the identity for the sine of a sum of angles.
equals sin   x   cos   x plus sin   x   cos   x Reverse the factors in the second term.
equals 2 sin   x   cos   x Simplify.

formula to know
Sine of a Double Angle
sin open parentheses 2 x close parentheses equals 2 sin   x   cos   x

Lastly, to find a double-angle identity for tangent, write tan open parentheses 2 x close parentheses equals tan open parentheses x plus x close parentheses.

tan open parentheses x plus x close parentheses equals fraction numerator tan   x plus tan   x over denominator 1 minus tan   x   tan   x end fraction Apply the identity for the tangent of a sum of angles.
equals fraction numerator 2 tan   x over denominator 1 minus tan squared x end fraction Simplify.

formula to know
Tangent of a Double Angle
tan open parentheses 2 x close parentheses equals fraction numerator 2 tan   x over denominator 1 minus tan squared x end fraction

EXAMPLE

Given sin   x equals 2 over 5 comma where straight pi over 2 less than x less than straight pi comma find the exact values of sin   2 x comma cos   2 x comma and tan   2 x.

First, note that sin open parentheses 2 x close parentheses equals 2 sin   x   cos   x. Since cos   x is not given but sin   x is known, use a Pythagorean identity to find cos   x.

Since straight pi over 2 less than x less than straight pi comma we know that cos   x less than 0.

sin squared x plus cos squared x equals 1 This is the Pythagorean identity.
open parentheses 2 over 5 close parentheses squared plus cos squared x equals 1 Replace sin   x with 2 over 5.
4 over 25 plus cos squared x equals 1 Simplify.
cos squared x equals 21 over 25 Subtract 4 over 25 from both sides.
cos   x equals short dash square root of 21 over 25 end root Apply the square root property. Since cos   x less than 0 comma only the negative solution is considered.
equals short dash fraction numerator square root of 21 over denominator 5 end fraction Simplify.

Thus, cos   x equals short dash fraction numerator square root of 21 over denominator 5 end fraction.

Then, sin open parentheses 2 x close parentheses equals 2 sin   x   cos   x equals 2 open parentheses 2 over 5 close parentheses open parentheses short dash fraction numerator square root of 21 over denominator 5 end fraction close parentheses equals short dash fraction numerator 4 square root of 21 over denominator 25 end fraction.

Next, find cos open parentheses 2 x close parentheses.

Since sin   x is given, it is most convenient to use the identity cos open parentheses 2 x close parentheses equals 1 minus 2 sin squared x.

Then, cos open parentheses 2 x close parentheses equals 1 minus 2 open parentheses 2 over 5 close parentheses squared equals 17 over 25.

Note: any version of the identities for cos open parentheses 2 x close parentheses could be used. You would get the same answer; it just might require more steps.

To find tan open parentheses 2 x close parentheses comma the double angle identity requires the value of tan   x to be known. While we could find this, remember also that tan open parentheses 2 x close parentheses equals fraction numerator sin open parentheses 2 x close parentheses over denominator cos open parentheses 2 x close parentheses end fraction comma and sin open parentheses 2 x close parentheses and cos open parentheses 2 x close parentheses have already been calculated.

Then, tan open parentheses 2 x close parentheses equals fraction numerator sin open parentheses 2 x close parentheses over denominator cos open parentheses 2 x close parentheses end fraction equals fraction numerator open parentheses begin display style fraction numerator short dash 4 square root of 21 over denominator 25 end fraction end style close parentheses over denominator open parentheses begin display style 17 over 25 end style close parentheses end fraction equals fraction numerator short dash 4 square root of 21 over denominator 17 end fraction.

try it
Given that cos   x equals short dash 21 over 29 comma where straight pi less than x less than fraction numerator 3 straight pi over denominator 2 end fraction.
Find the exact value of sin x.
sin squared x plus cos squared x equals 1 Use this identity since it relates sin   x and cos   x.
sin squared x plus open parentheses short dash 21 over 29 close parentheses squared equals 1 Replace cos   x with short dash 21 over 29.
sin squared x plus 441 over 841 equals 1 Simplify.
sin squared x equals 400 over 841 Isolate sin squared x to one side.
sin   x equals plus-or-minus square root of 400 over 841 end root equals plus-or-minus 20 over 29 Apply the square root principle.
sin   x equals short dash 20 over 29 Since x terminates in quadrant III, the value of sin   x is negative.
Find the exact value of sin(2x  ).
By the double angle identity, sin open parentheses 2 x close parentheses equals 2   sin   x   cos   x.

Since we now know the values of sin   x and cos   x comma we have:

sin open parentheses 2 x close parentheses equals 2 open parentheses short dash 20 over 29 close parentheses open parentheses short dash 21 over 29 close parentheses equals 840 over 841
Find the exact value of cos(2x  ).
There are three ways to find cos open parentheses 2 x close parentheses.

Since we have both sin   x and cos   x comma we’ll use cos open parentheses 2 x close parentheses equals cos squared x minus sin squared x.

Replacing the known values of sin   x and cos   x comma we have:

cos open parentheses 2 x close parentheses equals open parentheses short dash 21 over 29 close parentheses squared minus open parentheses short dash 20 over 29 close parentheses squared equals 41 over 841
Find the exact value of tan(2x  ).
Recall that tan open parentheses 2 x close parentheses equals fraction numerator sin open parentheses 2 x close parentheses over denominator cos open parentheses 2 x close parentheses end fraction.

Then, tan open parentheses 2 x close parentheses equals fraction numerator open parentheses 840 over 841 close parentheses over denominator open parentheses 41 over 841 close parentheses end fraction equals 840 over 41.

We can also use double-angle identities to prove trigonometric identities.

EXAMPLE

Prove the identity cos 2 x equals cos to the power of 4 x minus sin to the power of 4 x.

We’ll start with the right side.

cos to the power of 4 x minus sin to the power of 4 x equals open parentheses cos squared x plus sin squared x close parentheses open parentheses cos squared x minus sin squared x close parentheses Factor as a difference of squares.
equals open parentheses 1 close parentheses open parentheses cos squared x minus sin squared x close parentheses Apply the Pythagorean identity, cos squared x plus sin squared x equals 1.
equals cos 2 x Apply the cosine of a double angle identity, cos squared x minus sin squared x equals cos 2 x.

This proves the identity.

try it
Consider the equation open parentheses cos   t minus sin   t close parentheses squared equals 1 minus sin open parentheses 2 t close parentheses.
Prove that the equation is an identity.
Start with the left side, since it looks like it can be manipulated.

open parentheses cos   t minus sin   t close parentheses squared equals open parentheses cos   t minus sin   t close parentheses open parentheses cos   t minus sin   t close parentheses Rewrite as a product.
equals cos squared t minus 2   sin   t   cos   t plus sin squared t Expand the product.
equals 1 minus 2   sin   t   cos   t Use the identity sin squared t plus cos squared t equals 1.
equals 1 minus sin open parentheses 2 t close parentheses Replace 2   sin   t   cos   t with sin open parentheses 2 t close parentheses by way of the double angle identity. This identity is now verified since this is the right-hand side.

2. Using Power-Reducing Identities

Power-reducing identities are used to write even powers of sine, cosine, and tangent as trigonometric functions with smaller powers. These identities are very useful, particularly in calculus.

Recall that we had the identities cos open parentheses 2 x close parentheses equals 2 cos squared x minus 1 and cos open parentheses 2 x close parentheses equals 1 minus 2 sin squared x.

Solving the first equation for cos squared x gives cos squared x equals fraction numerator 1 plus cos open parentheses 2 x close parentheses over denominator 2 end fraction.

Solving the second equation for sin squared x gives sin squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction.

Also, since tan squared x equals fraction numerator sin squared x over denominator cos squared x end fraction comma we have tan squared x equals fraction numerator open parentheses fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction close parentheses over denominator open parentheses fraction numerator 1 plus cos open parentheses 2 x close parentheses over denominator 2 end fraction close parentheses end fraction equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 1 plus cos open parentheses 2 x close parentheses end fraction.

formula to know
Power-Reducing Identities
table attributes columnalign left end attributes row cell cos squared x equals fraction numerator 1 plus cos open parentheses 2 x close parentheses over denominator 2 end fraction end cell row cell sin squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction end cell row cell tan squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 1 plus cos open parentheses 2 x close parentheses end fraction end cell end table

EXAMPLE

Use power-reducing identities to write an equivalent expression for sin squared open parentheses 3 x close parentheses with no exponent larger than 1.

Applying the power-reducing formula sin squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction comma replace x with 3x.

Then, we have sin squared open parentheses 3 x close parentheses equals fraction numerator 1 minus cos open parentheses 2 times 3 x close parentheses over denominator 2 end fraction equals fraction numerator 1 minus cos open parentheses 6 x close parentheses over denominator 2 end fraction.

watch
Check out this video to see how to use power-reducing identities to rewrite sin to the power of 4 x as an expression with no powers.

EXAMPLE

Use power-reducing identities to write an equivalent expression for sin squared x   cos squared x with no exponent larger than 1.

sin squared x   cos squared x equals open parentheses fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction close parentheses open parentheses fraction numerator 1 plus cos open parentheses 2 x close parentheses over denominator 2 end fraction close parentheses Use the power-reducing identities.
equals fraction numerator 1 minus cos squared open parentheses 2 x close parentheses over denominator 4 end fraction Perform the multiplication and simplify.
equals fraction numerator sin squared open parentheses 2 x close parentheses over denominator 4 end fraction Apply a Pythagorean identity.
equals 1 fourth sin squared open parentheses 2 x close parentheses Rewrite with 1 fourth in front.
equals 1 fourth open parentheses fraction numerator 1 minus cos open parentheses 2 times 2 x close parentheses over denominator 2 end fraction close parentheses Apply a power-reducing identity.
equals 1 over 8 open parentheses 1 minus cos open parentheses 4 x close parentheses close parentheses Simplify.

Thus, sin squared x   cos squared x equals 1 over 8 open parentheses 1 minus cos open parentheses 4 x close parentheses close parentheses.

try it
Consider the expression cos to the power of 4 x equals cos squared x times cos squared x.
Apply the power-reducing identities once and expand the expression.
cos squared x times cos squared x equals open parentheses fraction numerator 1 plus cos   2 x over denominator 2 end fraction close parentheses open parentheses fraction numerator 1 plus cos   2 x over denominator 2 end fraction close parentheses Use power-reducing identities to replace the square terms.
equals fraction numerator open parentheses 1 plus cos   2 x close parentheses open parentheses 1 plus cos   2 x close parentheses over denominator 4 end fraction Write the product as a single fraction.
equals fraction numerator 1 plus 2   cos   2 x plus cos squared open parentheses 2 x close parentheses over denominator 4 end fraction Write the numerator in expanded form.
equals 1 fourth plus 1 half cos open parentheses 2 x close parentheses plus 1 fourth cos squared open parentheses 2 x close parentheses Perform the division by 4.

After applying the power-reducing identities once, cos squared x times cos squared x is equivalent to 1 fourth plus 1 half cos open parentheses 2 x close parentheses plus 1 fourth cos squared open parentheses 2 x close parentheses.
Apply a power-reducing identity on the last expression and simplify.
Notice that the last term in the previous answer has a square term. The power-reducing identity can be applied to this term as well.

By using the identity, cos squared open parentheses 2 x close parentheses equals fraction numerator 1 plus cos open parentheses 4 x close parentheses over denominator 2 end fraction.

Note that the angle on the right-hand side is 4 x comma since the original angle gets doubled when applying the power-reducing identity.

Now, replace this in the original expression and simplify:

equals 1 fourth plus 1 half cos open parentheses 2 x close parentheses plus 1 fourth open parentheses fraction numerator 1 plus cos open parentheses 4 x close parentheses over denominator 2 end fraction close parentheses Replace the square term with the power-reduced expression.
equals 1 fourth plus 1 half cos open parentheses 2 x close parentheses plus fraction numerator 1 plus cos open parentheses 4 x close parentheses over denominator 8 end fraction Perform the multiplication on the fractions.
equals 1 fourth plus 1 half cos open parentheses 2 x close parentheses plus 1 over 8 plus 1 over 8 cos open parentheses 4 x close parentheses Perform the division by 8.
equals 3 over 8 plus 1 half cos open parentheses 2 x close parentheses plus 1 over 8 cos open parentheses 4 x close parentheses Combine like terms open parentheses 1 fourth plus 1 over 8 equals 3 over 8 close parentheses.

Thus, the original expression, cos to the power of 4 x equals 3 over 8 plus 1 half cos open parentheses 2 x close parentheses plus 1 over 8 cos open parentheses 4 x close parentheses.

3. Using Half-Angle Identities

Bicycle ramps made for competition must vary in height depending on the skill level of the competitors. For advanced competitors at one particular event, the angle formed by the ramp and the ground should be the acute angle theta such that tan theta equals rise over run equals 5 over 3.

For novices at this same event, the angle is divided in half. Recalling that tan theta gives the steepness (slope) of a line that passes through the origin, what is the steepness of the ramp for novices? That is, what is the value of tan open parentheses theta over 2 close parentheses comma and can we get this value knowing that tan theta equals 5 over 3 ? We’ll be able to answer this question, and others, by establishing half-angle identities.

To start, consider the power-reducing identities:

cos squared x equals fraction numerator 1 plus cos open parentheses 2 x close parentheses over denominator 2 end fraction         sin squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction         tan squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 1 plus cos open parentheses 2 x close parentheses end fraction

In each identity, let theta equals 2 x comma then x equals theta over 2. We now have:

cos squared open parentheses theta over 2 close parentheses equals fraction numerator 1 plus cos theta over denominator 2 end fraction         sin squared open parentheses theta over 2 close parentheses equals fraction numerator 1 minus cos theta over denominator 2 end fraction         tan squared open parentheses theta over 2 close parentheses equals fraction numerator 1 minus cos theta over denominator 1 plus cos theta end fraction

Next, apply the square root principle to both sides of each equation:

cos open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 plus cos theta over denominator 2 end fraction end root         sin open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cos theta over denominator 2 end fraction end root         tan open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cos theta over denominator 1 plus cos theta end fraction end root

Note: the “plus-or-minus” is determined by the quadrant in which theta over 2 terminiates.

These identities collectively are known as the half-angle identities.

formula to know
Cosine of a Half-Angle
cos open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 plus cos theta over denominator 2 end fraction end root
Sine of a Half-Angle
sin open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cos theta over denominator 2 end fraction end root
Tangent of a Half-Angle (Three Versions)
tan open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cos theta over denominator 1 plus cos theta end fraction end root equals fraction numerator sin theta over denominator 1 plus cos theta end fraction equals fraction numerator 1 minus cos theta over denominator sin theta end fraction
Note: the “plus-or-minus” is determined by the quadrant in which theta over 2 terminates.

Just like the other identities we have used, the half-angle identities can be used to find exact values of angles that are half the value of a special angle.

EXAMPLE

Use a half-angle identity to find the exact value of sin open parentheses straight pi over 12 close parentheses.

Note that straight pi over 12 is equivalent to 15 degree.

sin open parentheses straight pi over 12 close parentheses equals sin 15 degree equals sin open parentheses fraction numerator 30 degree over denominator 2 end fraction close parentheses 15 degree is half of 30 degree comma and 30 degree is a special angle.
equals plus-or-minus square root of fraction numerator 1 minus cos 30 degree over denominator 2 end fraction end root Apply the appropriate half-angle identity.
equals square root of fraction numerator 1 minus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root Replace cos 30 degree with fraction numerator square root of 3 over denominator 2 end fraction.
Use the positive square root since 15 degree terminates in quadrant I, and all trigonometric functions have positive values in quadrant I.
equals square root of fraction numerator 2 minus square root of 3 over denominator 4 end fraction end root Inside the radical, multiply the numerator and denominator by 2.
equals fraction numerator square root of 2 minus square root of 3 end root over denominator 2 end fraction Take the square root of the numerator and denominator.

While this expression is very obscure, this is as far as we will simplify it. You may recall from earlier that we found that sin 15 degree equals fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction. While not obvious, both of these answers are equivalent.

watch
Have a look at this video where we will use half-angle identities to find the exact value of cos open parentheses theta over 2 close parentheses comma given cos theta equals short dash 5 over 6 comma where straight pi less than theta less than fraction numerator 3 straight pi over denominator 2 end fraction.

try it
Consider the identity tan open parentheses theta over 2 close parentheses equals fraction numerator 1 minus cos theta over denominator sin theta end fraction.
Use the identity to find the exact value of tan 22.5°.
To find tan open parentheses 22.5 degree close parentheses comma notice that 22.5 degree is half of 45 degree. Thus, we can use the half-angle identity with theta equals 45 degree.

tan open parentheses 22.5 degree close parentheses equals tan open parentheses fraction numerator 45 degree over denominator 2 end fraction close parentheses Write 22.5 degree equals fraction numerator 45 degree over denominator 2 end fraction to make use of the identity.
equals fraction numerator 1 minus cos   45 degree over denominator sin   45 degree end fraction Replace theta with 45 degree in the given formula.
equals fraction numerator 1 minus fraction numerator square root of 2 over denominator 2 end fraction over denominator fraction numerator square root of 2 over denominator 2 end fraction end fraction Replace the trigonometric functions with their exact values.
equals fraction numerator 2 minus square root of 2 over denominator square root of 2 end fraction Multiply by 2 over 2 to clear the complex fraction.
equals fraction numerator 2 square root of 2 minus 2 over denominator 2 end fraction Multiply by fraction numerator square root of 2 over denominator square root of 2 end fraction to rationalize the denominator.
equals square root of 2 minus 1 Perform the division by 2.

Therefore, tan open parentheses 22.5 degree close parentheses equals square root of 2 minus 1.

EXAMPLE

Given cos theta equals 3 over 5 comma where 270 degree less than theta less than 360 degree comma find the exact value of cos open parentheses theta over 2 close parentheses.

cos open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 plus cos theta over denominator 2 end fraction end root This is the half-angle identity.
equals short dash square root of fraction numerator 1 plus begin display style 3 over 5 end style over denominator 2 end fraction end root Since 270 degree less than theta less than 360 degree comma it follows that 135 degree less than theta over 2 less than 180 degree. Then, theta over 2 terminates in quadrant II, which means cos open parentheses theta over 2 close parentheses less than 0.
equals short dash square root of 4 over 5 end root Simplify under the radical.
equals short dash fraction numerator 2 over denominator square root of 5 end fraction Simplify the square root.
equals short dash fraction numerator 2 over denominator square root of 5 end fraction times fraction numerator square root of 5 over denominator square root of 5 end fraction equals short dash fraction numerator 2 square root of 5 over denominator 5 end fraction Rationalize the denominator.

Thus, cos open parentheses theta over 2 close parentheses equals short dash fraction numerator 2 square root of 5 over denominator 5 end fraction.

Now, let’s answer the question about the bicycle ramp from earlier.

EXAMPLE

Given tan theta equals 5 over 3 comma where theta is acute, find the exact value of tan open parentheses theta over 2 close parentheses.

Since all the identities for tan open parentheses theta over 2 close parentheses require us to know cos theta comma we’ll start by finding cos theta.

Since theta is acute, there are two methods that could be used:

  • Use a right triangle.
  • Use Pythagorean identities.
For this problem, we’ll use a right triangle, which is shown in the figure.

Given tan theta equals 5 over 3 comma the side opposite theta has length 5, and the side adjacent to theta has length 3.



Using the Pythagorean theorem, find the length of the hypotenuse.

a squared plus b squared equals c squared This is the Pythagorean theorem.
3 squared plus 5 squared equals c squared The legs are 3 and 5, so one of them is a and the other is b.
34 equals c squared Simplify.
c equals square root of 34 Apply the square root principle, Since c is the length of a side, only the positive solution is considered.

Thus, the length of the hypotenuse is square root of 34.

Then, it follows that cos theta equals fraction numerator 3 over denominator square root of 34 end fraction equals fraction numerator 3 square root of 34 over denominator 34 end fraction after rationalizing the denominator.

Similarly, sin theta equals fraction numerator 5 over denominator square root of 34 end fraction equals fraction numerator 5 square root of 34 over denominator 34 end fraction.

Next, use the identity tan open parentheses theta over 2 close parentheses equals fraction numerator 1 minus cos theta over denominator sin theta end fraction.

tan open parentheses theta over 2 close parentheses equals fraction numerator 1 minus cos theta over denominator sin theta end fraction This is a half-angle identity.
equals fraction numerator 1 minus begin display style fraction numerator 3 square root of 34 over denominator 34 end fraction end style over denominator open parentheses begin display style fraction numerator 5 square root of 34 over denominator 34 end fraction end style close parentheses end fraction Replace sin theta and cos theta with their values.
equals fraction numerator 34 minus 3 square root of 34 over denominator 5 square root of 34 end fraction Multiply the numerator and denominator by 34 to simplify the complex fraction.
equals fraction numerator 34 minus 3 square root of 34 over denominator 5 square root of 34 end fraction times fraction numerator square root of 34 over denominator square root of 34 end fraction equals fraction numerator 34 square root of 34 minus 102 over denominator 170 end fraction Rationalize the denominator.

Thus, tan open parentheses theta over 2 close parentheses equals fraction numerator 34 square root of 34 minus 102 over denominator 170 end fraction. For context, this is approximately 0.566, which is approximately one third of the steepness of the original ramp whose slope is 5 over 3.

try it
Find the exact value of tan open parentheses theta over 2 close parentheses given cos theta equals 1 fifth comma where fraction numerator 3 straight pi over denominator 2 end fraction less than theta less than 2 straight pi.
What is the exact value?
Recall that tan open parentheses theta over 2 close parentheses equals fraction numerator 1 minus cos   theta over denominator sin   theta end fraction. We are given the value of cos   theta comma which is used to find the exact value of sin   theta.

sin squared theta plus cos squared theta equals 1 This identity relates sin   theta and cos   theta.
sin squared theta plus open parentheses 1 fifth close parentheses squared equals 1 Replace cos   theta with 1 fifth.
sin squared theta plus 1 over 25 equals 1 Simplify.
sin squared theta equals 24 over 25 Isolate the square term to one side.
sin   theta equals plus-or-minus square root of 24 over 25 end root equals plus-or-minus fraction numerator 2 square root of 6 over denominator 5 end fraction Apply the square root principle, simplify the radical expression.
sin   theta equals short dash fraction numerator 2 square root of 6 over denominator 5 end fraction Since theta terminates in quadrant IV, sin   theta is negative.

Next, use the values of cos   theta and sin   theta to find tan open parentheses theta over 2 close parentheses.

tan open parentheses theta over 2 close parentheses equals fraction numerator 1 minus cos   theta over denominator sin   theta end fraction The half-angle identity.
equals fraction numerator 1 minus 1 fifth over denominator open parentheses short dash fraction numerator 2 square root of 6 over denominator 5 end fraction close parentheses end fraction Replace the trigonometric expressions with their values.
equals fraction numerator 5 minus 1 over denominator short dash 2 square root of 6 end fraction equals fraction numerator 4 over denominator short dash 2 square root of 6 end fraction equals short dash fraction numerator 2 over denominator square root of 6 end fraction Multiply by 5 over 5 to clear the complex fraction, then simplify.
equals short dash fraction numerator 2 square root of 6 over denominator 6 end fraction equals short dash fraction numerator square root of 6 over denominator 3 end fraction Multiply by fraction numerator square root of 6 over denominator square root of 6 end fraction to rationalize the denominator, then simplify.

Therefore, tan open parentheses theta over 2 close parentheses equals short dash fraction numerator square root of 6 over denominator 3 end fraction.

summary
In this lesson, you learned that by using the sum of angles identities, you can establish double angle identities that give values of cos open parentheses 2 x close parentheses comma sin open parentheses 2 x close parentheses comma and tan open parentheses 2 x close parentheses in terms of trigonometric functions of x. You can also use double-angle identities to prove trigonometric identities. You also learned how to use power-reducing identities to write even powers of sine, cosine, and tangent as trigonometric functions with smaller powers. Finally, you learned how to use half-angle identities to find exact values of angles that are half the value of a special angle. In summary, double-angle identities, power-reducing identities, and half-angle identities all are used in conjunction with other identities to evaluate expressions, simplify expressions, and verify trigonometric identities.

SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS

Formulas to Know
Cosine of a Double Angle

cos open parentheses 2 x close parentheses equals cos squared x minus sin squared x</p> <p>cos open parentheses 2 x close parentheses equals 1 minus 2 sin squared x</p> <p>cos open parentheses 2 x close parentheses equals 2 cos squared x minus 1

Cosine of a Half-Angle

cos open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 plus cos theta over denominator 2 end fraction end root

Power-Reducing Identities

cos squared x equals fraction numerator 1 plus cos open parentheses 2 x close parentheses over denominator 2 end fraction</p> <p>sin squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 2 end fraction</p> <p>tan squared x equals fraction numerator 1 minus cos open parentheses 2 x close parentheses over denominator 1 plus cos open parentheses 2 x close parentheses end fraction

Sine of a Double Angle

sin open parentheses 2 x close parentheses equals 2 sin   x   cos   x

Sine of a Half-Angle

sin open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cos theta over denominator 2 end fraction end root

Tangent of a Double Angle

tan open parentheses 2 x close parentheses equals fraction numerator 2 tan   x over denominator 1 minus tan squared x end fraction

Tangent of a Half-Angle (Three Versions)

tan open parentheses theta over 2 close parentheses equals plus-or-minus square root of fraction numerator 1 minus cos theta over denominator 1 plus cos theta end fraction end root equals fraction numerator sin theta over denominator 1 plus cos theta end fraction equals fraction numerator 1 minus cos theta over denominator sin theta end fraction

Note: the “plus-or-minus” is determined by the quadrant in which theta over 2 terminates.

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