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Sum and Difference Identities

Author: Sophia
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what's covered
In this lesson, you will use identities that will allow you to evaluate trigonometric expressions for angles that are sums and differences of other angles. Specifically, this lesson will cover:

Table of Contents

1. Evaluating the Cosine of a Sum and Difference of Angles

We now look at identities which we can use to evaluate trigonometric expressions when the angle is a sum or difference of two other angles.

If A and B are angles, then we have the following identities.

formula to know
Cosine of a Sum or Difference of Angles
table attributes columnalign left end attributes row cell cos open parentheses A plus B close parentheses equals cos   A   cos   B minus sin   A   sin   B end cell row cell cos open parentheses A minus B close parentheses equals cos   A   cos   B plus sin   A   sin   B end cell end table

One way to use these identities is to find exact values of cosines of sums and differences of special angles.

EXAMPLE

Note that 75 degree equals 30 degree plus 45 degree. Use this fact to find cos 75 degree.

cos open parentheses A plus B close parentheses equals cos   A   cos   B minus sin   A   sin   B This is the identity used.
cos open parentheses 30 degree plus 45 degree close parentheses equals cos 30 degree cos 45 degree minus sin 30 degree sin 45 degree Replace A with 30 degree and B with 45 degree.
cos 75 degree equals open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses minus open parentheses 1 half close parentheses open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses Replace all trigonometric functions with their exact values.
cos 75 degree equals fraction numerator square root of 6 over denominator 4 end fraction minus fraction numerator square root of 2 over denominator 4 end fraction Simplify.
cos 75 degree equals fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction Write the result as a single fraction.

Thus, cos 75 degree equals fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction. Note that this exact value is quite more complicated than the exact values we’ve learned for our special angles. While it’s useful to be able to find an exact value for cos 75 degree comma it is not very useful to memorize it.

Note also that 75 degree is equal to fraction numerator 5 straight pi over denominator 12 end fraction radian. This means that cos open parentheses fraction numerator 5 straight pi over denominator 12 end fraction close parentheses equals fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction.

try it
Consider the expression cos 15 degree.
Use the cosine of a difference formula to find the exact value of cos(15°).
First, identify two special angles whose difference is 15 degree colon 15 degree equals 45 degree minus 30 degree

Next, replace 15 degree with 45 degree minus 30 degree comma then apply the difference formula.

cos 15 degree equals cos open parentheses 45 degree minus 30 degree close parentheses Replace 15 degree with 45 degree minus 30 degree.
equals cos   45 degree cos   30 degree plus sin   45 degree sin   30 degree Use the identity cos open parentheses A minus B close parentheses equals cos   A   cos   B plus sin   A   sin   B.
equals open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses plus open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses open parentheses 1 half close parentheses Replace each trig function with its value.
equals fraction numerator square root of 6 over denominator 4 end fraction plus fraction numerator square root of 2 over denominator 4 end fraction Perform each multiplication.
equals fraction numerator square root of 6 plus square root of 2 over denominator 4 end fraction Write as a single fraction.

Thus, the exact value of cos 15 degree is fraction numerator square root of 6 plus square root of 2 over denominator 4 end fraction.

watch
In this video, we will find the exact value of cos open parentheses x minus y close parentheses comma where values of cos   x and cos   y are known.

EXAMPLE

Given cos   alpha equals short dash 3 over 5 comma where straight pi over 2 less than alpha less than straight pi comma and sin beta equals 5 over 13 comma where 0 less than beta less than straight pi over 2 comma find the exact value of cos open parentheses alpha plus beta close parentheses.

By an identity stated above, cos open parentheses alpha plus beta close parentheses equals cos   alpha cos beta minus sin   alpha sin beta.

We were given the values of cos   alpha and sin beta comma but not sin   alpha and cos beta. These can be found using Pythagorean identities.

First, find cos beta. Since 0 less than beta less than straight pi over 2 comma we know cos beta greater than 0.

sin squared beta plus cos squared beta equals 1 This is the identity to use when given sin beta and we want to find cos beta.
open parentheses 5 over 13 close parentheses squared plus cos squared beta equals 1 Replace sin beta with 5 over 13.
25 over 169 plus cos squared beta equals 1 Simplify.
cos squared beta equals 144 over 169 Subtract 25 over 169 from both sides.
cos beta equals 12 over 13 Apply the square root principle. Since cos beta greater than 0 for a quadrant I angle, only the positive solution is considered.

Thus, cos beta equals 12 over 13.

Next, find sin   alpha. Since straight pi over 2 less than alpha less than straight pi comma we know sin   alpha greater than 0.

sin squared alpha plus cos squared alpha equals 1 This is the identity to use when given cos alpha and we want to find sin alpha.
sin squared alpha plus open parentheses short dash 3 over 5 close parentheses squared equals 1 Replace cos alpha with short dash 3 over 5.
sin squared alpha plus 9 over 25 equals 1 Simplify.
sin squared alpha equals 16 over 25 Subtract 9 over 25 from both sides.
sin   alpha equals 4 over 5 Apply the square root principle. Since sin alpha greater than 0 for a quadrant I angle, only the positive solution is considered.

Thus, sin   alpha equals 4 over 5.

Now, evaluate cos open parentheses alpha plus beta close parentheses.

cos open parentheses alpha plus beta close parentheses equals cos   alpha cos beta minus sin   alpha sin beta This is the sum of angles identity.
equals open parentheses short dash 3 over 5 close parentheses open parentheses 12 over 13 close parentheses minus open parentheses 4 over 5 close parentheses open parentheses 5 over 13 close parentheses Replace all trigonometric functions of alpha and beta with their values.
equals short dash 36 over 65 minus 20 over 65
equals short dash 56 over 65 		Simplify.

Thus, cos open parentheses alpha plus beta close parentheses equals short dash 56 over 65.

try it
Given sin   x equals 40 over 41 and sin   y equals 24 over 25 comma where both x and y are acute angles.
Find the exact value of cos(x - y  ).
First, note that cos open parentheses x minus y close parentheses equals cos   x   cos   y plus sin   x   sin   y.

We are given sin   x equals 40 over 41 comma which can be used to find cos   x.

sin squared x plus cos squared x equals 1 This identity relates sin   x and cos   x.
open parentheses 40 over 41 close parentheses squared plus cos squared x equals 1 Replace sin   x with 40 over 41.
1600 over 1681 plus cos squared x equals 1 Simplify.
cos squared x equals 81 over 1681 Isolate cos squared x.
cos   x equals square root of 81 over 1681 end root equals 9 over 41 Apply the square root principle to both sides. Since x is an acute angle, it terminates in quadrant I, which means all trigonometric ratios are positive.

Similarly, we are also given sin   y equals 24 over 25 comma which can be used to find cos   y.

sin squared y plus cos squared y equals 1 This identity relates sin   y and cos   y.
open parentheses 24 over 25 close parentheses squared plus cos squared y equals 1 Replace sin   y with 24 over 25.
576 over 625 plus cos squared y equals 1 Simplify.
cos squared y equals 49 over 625 Isolate cos squared y.
cos   y equals square root of 49 over 625 end root equals 7 over 25 Apply the square root principle to both sides. Since y is an acute angle, it terminates in quadrant I, which means all trigonometric ratios are positive.

Now that we have the four trigonometric ratios we need, we can evaluate cos open parentheses x minus y close parentheses.

cos open parentheses x minus y close parentheses equals cos   x   cos   y plus sin   x   sin   y</dd></dl></dd></dl> space space space space space space space space space space space space space space space space space space space equals open parentheses 9 over 41 close parentheses open parentheses 7 over 25 close parentheses plus open parentheses 40 over 41 close parentheses open parentheses 24 over 25 close parentheses space space space space space space space space space space space space space space space space space space space equals 63 over 1025 plus 960 over 1025 space space space space space space space space space space space space space space space space space space space equals 1023 over 1025

Sum and difference formulas can also be used to simplify expressions.

EXAMPLE

Use an appropriate identity to simplify cos open parentheses straight pi minus x close parentheses.

cos open parentheses straight pi minus x close parentheses equals cosπ   cos   x plus sinπ   sin   x Use the cosine of a difference of angles identity.
equals short dash 1 open parentheses cos   x close parentheses plus 0 open parentheses sin   x close parentheses Replace cosπ and sinπ with their values.
equals short dash cos   x Simplify.

Therefore, cos open parentheses straight pi minus x close parentheses equals short dash cos   x.

watch
In the following video, we’ll use the sum formula to find the exact value of cos open parentheses fraction numerator 2 straight pi over denominator 3 end fraction plus fraction numerator 3 straight pi over denominator 4 end fraction close parentheses.

try it
Consider the expression cos open parentheses x plus straight pi over 3 close parentheses.
Use the sum of angles identity to expand the expression.
Using the sum of angles identity, we have the following:

cos open parentheses x plus straight pi over 3 close parentheses equals cos   x   cos straight pi over 3 minus sin   x   sin straight pi over 3 Use the identity cos open parentheses A plus B close parentheses equals cos   A   cos   B minus sin   A   sin   B.
equals cos   x open parentheses 1 half close parentheses minus sin   x open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses Replace cos straight pi over 3 and sin straight pi over 3 with their exact values.
equals 1 half cos   x minus fraction numerator square root of 3 over denominator 2 end fraction sin   x Rewrite numerical coefficients before trigonometric functions.

Therefore, cos open parentheses x plus straight pi over 3 close parentheses can be written as 1 half cos   x minus fraction numerator square root of 3 over denominator 2 end fraction sin   x.

2. Evaluating the Sine of a Sum and Difference of Angles

Similar to the cosine formulas, we have the following:

formula to know
Sine of a Sum or Difference of Angles
table attributes columnalign left end attributes row cell sin open parentheses A plus B close parentheses equals sin   A   cos   B plus cos   A   sin   B end cell row cell sin open parentheses A minus B close parentheses equals sin   A   cos   B minus cos   A   sin   B end cell end table

Let’s use these identities to find exact values.

EXAMPLE

Find the exact value of sin open parentheses straight pi over 12 close parentheses.

Note that straight pi over 12 is equal to 15 degree comma which is 45 degree minus 30 degree. Thus, using radians, we write straight pi over 12 equals straight pi over 4 minus straight pi over 6.

sin open parentheses A minus B close parentheses equals sin   A   cos   B minus cos   A   sin   B This is the identity we will use.
sin open parentheses straight pi over 4 minus straight pi over 6 close parentheses equals sin straight pi over 4 cos straight pi over 6 minus cos straight pi over 4 sin straight pi over 6 Replace A with straight pi over 4 and B with straight pi over 6.
sin open parentheses straight pi over 12 close parentheses equals fraction numerator square root of 2 over denominator 2 end fraction times fraction numerator square root of 3 over denominator 2 end fraction minus fraction numerator square root of 2 over denominator 2 end fraction times 1 half Find the exact value of each trigonometric function.
sin open parentheses straight pi over 12 close parentheses equals fraction numerator square root of 6 over denominator 4 end fraction minus fraction numerator square root of 2 over denominator 4 end fraction Simplify.
sin open parentheses straight pi over 12 close parentheses equals fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction Write as a single fraction.

Thus, sin open parentheses straight pi over 12 close parentheses equals fraction numerator square root of 6 minus square root of 2 over denominator 4 end fraction.

try it
Consider the expression sin open parentheses 105 degree close parentheses.
Use the sine of a sum formula to find the exact value of the expression.
First, identify two special angles whose sum is 105 degree colon 105 degree equals 45 degree plus 60 degree

Next, replace 105 degree with 45 degree plus 60 degree comma then apply the sum formula.

sin   105 degree equals sin open parentheses 45 degree plus 60 degree close parentheses Replace 105 degree with 45 degree plus 60 degree.
equals sin   45 degree   cos   60 degree plus sin   60 degree   cos   45 degree Use the identity sin open parentheses A plus B close parentheses equals sin   A   cos   B plus sin   B   cos   A.
equals open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses open parentheses 1 half close parentheses plus open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses Replace each trig function with its value.
equals fraction numerator square root of 2 over denominator 4 end fraction plus fraction numerator square root of 6 over denominator 4 end fraction Perform each multiplication.
equals fraction numerator square root of 2 plus square root of 6 over denominator 4 end fraction Write as a single fraction.

Thus, the exact value of sin   105 degree is fraction numerator square root of 2 plus square root of 6 over denominator 4 end fraction.

Now, let’s simplify expressions.

EXAMPLE

Use an appropriate identity to expand the expression sin open parentheses x plus fraction numerator 3 straight pi over denominator 2 end fraction close parentheses.

sin open parentheses x plus fraction numerator 3 straight pi over denominator 2 end fraction close parentheses This is the expression we are going to expand.
equals sin   x   cos open parentheses fraction numerator 3 straight pi over denominator 2 end fraction close parentheses plus cos   x   sin open parentheses fraction numerator 3 straight pi over denominator 2 end fraction close parentheses Apply the sine of a sum of angles identity.
equals sin   x open parentheses 0 close parentheses plus cos   x open parentheses short dash 1 close parentheses cos open parentheses fraction numerator 3 straight pi over denominator 2 end fraction close parentheses equals 0 and sin open parentheses fraction numerator 3 straight pi over denominator 2 end fraction close parentheses equals short dash 1
equals short dash cos   x Simplify.

Thus, sin open parentheses x plus fraction numerator 3 straight pi over denominator 2 end fraction close parentheses equals short dash cos   x.

try it
Consider the expression sin open parentheses straight pi minus x close parentheses.
Expand the expression by using an appropriate identity.
Apply the difference of angles identity, then simplify:

sin open parentheses straight pi minus x close parentheses equals sin   straight pi   cos   x minus sin   x   cos   straight pi Apply the identity sin open parentheses A minus B close parentheses equals sin   A   cos   B minus sin   B   cos   A.
equals open parentheses 0 close parentheses cos   x minus sin   x open parentheses short dash 1 close parentheses Replace sin   straight pi and cos   straight pi with their exact values.
equals sin   x The first term is 0, simplify the second term.

This means that sin open parentheses straight pi minus x close parentheses is equivalent to sin   x.

watch
Have a look at this video where we will find the exact value of sin open parentheses x plus y close parentheses when given values of other trigonometric functions of x and y.

3. Evaluating the Tangent of a Sum and Difference of Angles

formula to know
Tangent of a Sum or Difference of Angles
table attributes columnalign left end attributes row cell tan open parentheses A plus B close parentheses equals fraction numerator tan   A plus tan   B over denominator 1 minus tan   A   tan   B end fraction end cell row cell tan open parentheses A minus B close parentheses equals fraction numerator tan   A minus tan   B over denominator 1 plus tan   A   tan   B end fraction end cell end table

EXAMPLE

Find the exact value of tan open parentheses 15 degree close parentheses.

This time, we'll use 15 degree equals 60 degree minus 45 degree.

tan open parentheses A minus B close parentheses equals fraction numerator tan   A minus tan   B over denominator 1 plus tan   A   tan   B end fraction This is the identity that is used.
tan open parentheses 60 degree minus 45 degree close parentheses equals fraction numerator tan 60 degree minus tan 45 degree over denominator 1 plus tan 60 degree tan 45 degree end fraction Replace A with 60 degree and B with 45 degree.
tan open parentheses 15 degree close parentheses equals fraction numerator square root of 3 minus 1 over denominator 1 plus square root of 3 open parentheses 1 close parentheses end fraction Replace tan open parentheses 60 degree close parentheses and tan open parentheses 45 degree close parentheses with their exact values.
equals fraction numerator square root of 3 minus 1 over denominator 1 plus square root of 3 end fraction Simplify.
equals fraction numerator square root of 3 minus 1 over denominator 1 plus square root of 3 end fraction times fraction numerator open parentheses 1 minus square root of 3 close parentheses over denominator open parentheses 1 minus square root of 3 close parentheses end fraction Rationalize the denominator.
equals fraction numerator 2 square root of 3 minus 4 over denominator short dash 2 end fraction Perform the multiplications.
equals 2 minus square root of 3 Perform the division; write the positive term first.

Thus, tan open parentheses 15 degree close parentheses equals 2 minus square root of 3.

try it
Consider the expression tan open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses.
Use an appropriate identity to find the exact value of the expression.
First, note that fraction numerator 7 straight pi over denominator 12 end fraction radians is equivalent to fraction numerator 7 straight pi over denominator 12 end fraction times 180 over straight pi equals 105 degree. Therefore, we first seek two angles whose sum is 105 degree. The most convenient choices are 60 degree and 45 degree.

Next, use the sum of angles identity to find tan open parentheses 105 degree close parentheses colon

tan open parentheses 105 degree close parentheses equals tan open parentheses 60 degree plus 45 degree close parentheses Replace 105 degree with 60 degree plus 45 degree.
equals fraction numerator tan   60 degree plus tan   45 degree over denominator 1 minus tan   60 degree   tan   45 degree end fraction Use the identity tan open parentheses A plus B close parentheses equals fraction numerator tan   A plus tan   B over denominator 1 minus tan   A   tan   B end fraction.
equals fraction numerator square root of 3 plus 1 over denominator 1 minus square root of 3 open parentheses 1 close parentheses end fraction Replace tan open parentheses 60 degree close parentheses and tan open parentheses 45 degree close parentheses with their exact values.
equals fraction numerator square root of 3 plus 1 over denominator 1 minus square root of 3 end fraction Simplify.
equals fraction numerator square root of 3 plus 1 over denominator 1 minus square root of 3 end fraction times fraction numerator 1 plus square root of 3 over denominator 1 plus square root of 3 end fraction Multiply numerator and denominator by 1 plus square root of 3 comma the conjugate of the denominator.
equals fraction numerator open parentheses square root of 3 plus 1 close parentheses open parentheses 1 plus square root of 3 close parentheses over denominator open parentheses 1 minus square root of 3 close parentheses open parentheses 1 plus square root of 3 close parentheses end fraction Write as a single fraction with appropriate grouping symbols.
equals fraction numerator 1 plus 2 square root of 3 plus 3 over denominator 1 minus 3 end fraction Expand the numerator and denominator.
equals fraction numerator 4 plus 2 square root of 3 over denominator short dash 2 end fraction Simplify.
equals short dash 2 minus square root of 3 Perform the division.

Thus, the exact value of tan open parentheses fraction numerator 7 straight pi over denominator 12 end fraction close parentheses is short dash 2 minus square root of 3.

Note: The value is negative, which makes sense since the angle fraction numerator 7 straight pi over denominator 12 end fraction terminates in quadrant II, where tangent values are negative.

EXAMPLE

Given tan   alpha equals 1 fifth and tan   beta equals short dash 2 over 3 comma find the exact value of tan open parentheses alpha minus beta close parentheses.

tan open parentheses alpha minus beta close parentheses equals fraction numerator tan   alpha minus tan   beta over denominator 1 plus tan   alpha   tan   beta end fraction This is the difference of angles identity for tangent.
equals fraction numerator begin display style 1 fifth end style minus open parentheses short dash begin display style 2 over 3 end style close parentheses over denominator 1 plus open parentheses 1 fifth close parentheses open parentheses short dash 2 over 3 close parentheses end fraction tan   alpha equals 1 fifth and tan   beta equals short dash 2 over 3
equals fraction numerator open parentheses begin display style 13 over 15 end style close parentheses over denominator open parentheses 1 minus begin display style 2 over 15 end style close parentheses end fraction Simplify the numerator and denominator.
equals 1 Simplify.

Thus, tan open parentheses alpha minus beta close parentheses equals 1.

Notice that we didn’t need quadrant information for the angles in this problem. That is because we were given values of the tangent function, and that is all that was required to use in the difference of angles identity.

try it
Given sin   alpha equals short dash 4 over 5 and tan   beta equals 3 comma where straight pi less than alpha less than fraction numerator 3 straight pi over denominator 2 end fraction.
Find the value of cos  α.
Given sin   alpha comma find cos   alpha by using an appropriate identity:

sin squared alpha plus cos squared alpha equals 1 Use this identity since it relates sin   alpha and cos   alpha.
open parentheses short dash 4 over 5 close parentheses squared plus cos squared alpha equals 1 Replace sin   alpha with short dash 4 over 5.
16 over 25 plus cos squared alpha equals 1 Simplify.
cos squared alpha equals 9 over 25 Isolate cos squared alpha to one side.
cos   alpha equals plus-or-minus square root of 9 over 25 end root equals plus-or-minus 3 over 5 Apply the square root principle.
cos   alpha equals short dash 3 over 5 Since alpha terminates in quadrant III, the value of cos   alpha is negative.
Find the value of tan  α.
tan   alpha equals fraction numerator sin   alpha over denominator cos   alpha end fraction equals fraction numerator open parentheses short dash 4 over 5 close parentheses over denominator open parentheses short dash 3 over 5 close parentheses end fraction equals 4 over 3
Find the exact value of tan(α + β).
To find tan open parentheses alpha plus beta close parentheses comma use the sum of angles identity, then evaluate.

tan open parentheses alpha plus beta close parentheses equals fraction numerator tan   alpha plus tan   beta over denominator 1 minus tan   alpha   tan   beta end fraction</dd></dl></dd></dl> space space space space space space space space space space space space space space space space space space space equals fraction numerator 4 over 3 plus 3 over denominator 1 minus open parentheses 4 over 3 close parentheses open parentheses 3 close parentheses end fraction space space space space space space space space space space space space space space space space space space space equals fraction numerator open parentheses 13 over 3 close parentheses over denominator short dash 3 end fraction space space space space space space space space space space space space space space space space space space space equals short dash 13 over 9

EXAMPLE

Use an appropriate identity to write an expanded expression for tan open parentheses x plus straight pi over 4 close parentheses.

tan open parentheses A plus B close parentheses equals fraction numerator tan   A plus tan   B over denominator 1 minus tan   A   tan   B end fraction This is the identity that is used.
tan open parentheses x plus straight pi over 4 close parentheses equals fraction numerator tan   x plus tan straight pi over 4 over denominator 1 minus tan   x   tan straight pi over 4 end fraction A equals x and B equals straight pi over 4
equals fraction numerator tan   x plus 1 over denominator 1 minus tan   x open parentheses 1 close parentheses end fraction tan open parentheses straight pi over 4 close parentheses equals 1
equals fraction numerator tan   x plus 1 over denominator 1 minus tan   x end fraction Simplify.

Thus, tan open parentheses x plus straight pi over 4 close parentheses equals fraction numerator tan   x plus 1 over denominator 1 minus tan   x end fraction comma which could also be written as fraction numerator 1 plus tan   x over denominator 1 minus tan   x end fraction.

try it
Consider the expression tan open parentheses straight pi minus theta close parentheses.
Apply an appropriate identity to expand and simplify this expression.
By using the difference of angles identity, we have the following:

tan open parentheses straight pi minus theta close parentheses equals fraction numerator tan   straight pi minus tan   theta over denominator 1 plus tan   straight pi   tan   theta end fraction Use the identity tan open parentheses A minus B close parentheses equals fraction numerator tan   A minus tan   B over denominator 1 plus tan   A   tan   B end fraction.
equals fraction numerator 0 minus tan   theta over denominator 1 plus open parentheses 0 close parentheses tan   theta end fraction tan   straight pi equals 0
equals short dash fraction numerator tan   theta over denominator 1 end fraction Simplify.
equals short dash tan theta Simplify.

Therefore, the expression tan open parentheses straight pi minus theta close parentheses simplifies to short dash tan theta.

4. Verifying Trigonometric Identities With Sum and Difference Formulas

We apply the same concepts that we did earlier. The only new challenge is that sum and difference identities may be used.

EXAMPLE

Verify that the equation tan   A plus cot   B equals fraction numerator cos open parentheses A minus B close parentheses over denominator cos   A   sin   B end fraction.

fraction numerator cos open parentheses A minus B close parentheses over denominator cos   A   sin   B end fraction Start with the right side since it can be manipulated; the left side is simplified.
equals fraction numerator cos   A cos   B plus sin   A sin   B over denominator cos   A sin   B end fraction Use the cosine of a difference of angles identity in the numerator.
equals fraction numerator cos   A cos   B over denominator cos   A sin   B end fraction plus fraction numerator sin   A sin   B over denominator cos   A sin   B end fraction Split the expression into single fractions.
equals fraction numerator cos   B over denominator sin   B end fraction plus fraction numerator sin   A over denominator cos   A end fraction Cancel common factors in each fraction.
equals cot   B plus tan   A Use the quotient identities.
equals tan   A plus cot   B Addition is commutative; rewrite in order to match the left side of the original equation.

This verifies the identity.

try it
Consider the equation sin open parentheses x plus y close parentheses minus sin open parentheses x minus y close parentheses equals 2   sin   y   cos   x.
Verity that the equation is an identity.
Start with the left side, since identities can be applied.

sin open parentheses x plus y close parentheses minus sin open parentheses x minus y close parentheses This is the left side of the equation.
equals sin   x   cos   y plus sin   y   cos   x minus open parentheses sin   x   cos   y minus sin   y   cos   x close parentheses Apply the sum of angles identity to the first expression and the difference of angles identity on the second term. Note the use of parentheses.
equals sin   x   cos   y plus sin   y   cos   x minus sin   x   cos   y plus sin   y   cos   x Distribute.
equals 2   sin   y   cos   x Combine like terms. This verifies the identity since this is the right-hand side of the equation.

summary
In this lesson, you examined several identities that can be used to evaluate trigonometric expressions when the angle is a sum or difference of two other angles, learning how to apply the formulas for the cosine of a sum or difference of angles, the sine of a sum and difference of angles, and the tangent of a sum and difference of angles. You also learned how to use these sum and difference formulas to verify trigonometric identities, applying the same concepts you learned in the last tutorial.

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 Sum or Difference of Angles

cos open parentheses A plus B close parentheses equals cos   A   cos   B minus sin   A   sin   B</p> <p>cos open parentheses A minus B close parentheses equals cos   A   cos   B plus sin   A   sin   B

Sine of a Sum or Difference of Angles

sin open parentheses A plus B close parentheses equals sin A cos B plus cos A sin B sin left parenthesis A minus B right parenthesis equals sin A cos B minus cos A sin B

Tangent of a Sum or Difference of Angles

tan open parentheses A plus B close parentheses equals fraction numerator tan   A plus tan   B over denominator 1 minus tan   A   tan   B end fraction</p> <p>tan open parentheses A minus B close parentheses equals fraction numerator tan   A minus tan   B over denominator 1 plus tan   A   tan   B end fraction

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