3.4: Sum and Difference Identities
Learning Objectives
 Use and identify the sum and difference identities.
 Apply the sum and difference identities to solve trigonometric equations.
 Find the exact value of a trigonometric function for certain angles.
In this section we are going to explore , and . These identities have very useful expansions and can help to solve identities and equations.
Sum and Difference Formulas: Cosine
Is ? Upon appearance, yes, it is. This section explores how to find an expression that would equal . To simplify this, let the two given angles be and where .
Begin with the unit circle and place the angles and in standard position as shown in Figure A. Point Pt1 lies on the terminal side of , so its coordinates are and Point Pt2 lies on the terminal side of so its coordinates are . Place the in standard position, as shown in Figure B. The point A has coordinates and the Pt3 is on the terminal side of the angle , so its coordinates are .
Triangles in figure A and Triangle in figure B are congruent. (Two sides and the included angle, , are equal). Therefore the unknown side of each triangle must also be equal. That is:
Applying the distance formula to the triangles in Figures A and B and setting them equal to each other:
Square both sides to eliminate the square root.
FOIL all four squared expressions and simplify.
In , the difference formula for cosine, you can substitute to obtain: or . since and , then , which is the sum formula for cosine.
Using the Sum and Difference Identities of Cosine
The sum/difference formulas for cosine can be used to establish other identities:
Example 1: Find an equivalent form of using the cosine difference formula.
Solution:
We know that is a true identity because of our understanding of the sine and cosine curves, which are a phase shift of off from each other.
The cosine formulas can also be used to find exact values of cosine that we weren’t able to find before, such as , among others.
Example 2: Find the exact value of
Solution: Use the difference formula where and .
Example 3: Find the exact value of .
Solution: There may be more than one pair of key angles that can add up (or subtract to) . Both pairs, and , will yield the correct answer.
1.
2.
You do not need to do the problem multiple ways, just the one that seems easiest to you.
Example 4: Find the exact value of , in radians.
Solution: , notice that and
Sum and Difference Identities: Sine
To find , use Example 1, from above:
In conclusion, , which is the sum formula for sine.
To obtain the identity for :
In conclusion, , so, this is the difference formula for sine.
Example 5: Find the exact value of
Solution: Recall that there are multiple angles that add or subtract to equal any angle. Choose whichever formula that you feel more comfortable with.
Example 6: Given , where is in Quadrant II, and , where is in Quadrant I, find the exact value of .
Solution: To find the exact value of , here we use . The values of and are known, however the values of and need to be found.
Use , to find the values of each of the missing cosine values.
For , substituting transforms to or , however, since is in Quadrant II, the cosine is negative, .
For use and substitute or and and since is in Quadrant I,
Now the sum formula for the sine of two angles can be found:
Sum and Difference Identities: Tangent
To find the sum formula for tangent:
In conclusion, . Substituting for in the above results in the difference formula for tangent:
Example 7: Find the exact value of .
Solution: Use the difference formula for tangent, with
To verify this on the calculator, and .
Using the Sum and Difference Identities to Verify Other Identities
Example 8: Verify the identity
Example 9: Show
Solution: First, expand left hand side using the sum and difference formulas:
Solving Equations with the Sum and Difference Formulas
Just like the section before, we can incorporate all of the sum and difference formulas into equations and solve for values of . In general, you will apply the formula before solving for the variable. Typically, the goal will be to isolate , or and then apply the inverse. Remember, that you may have to use the identities in addition to the formulas seen in this section to solve an equation.
Example 10: Solve in the interval .
Solution: First, get by itself, by dividing both sides by .
Now, expand the left side using the sine difference formula.
The when is .
Example 11: Find all the solutions for in the interval .
Solution: Get the by itself and then take the square root.
Now, use the cosine sum formula to expand and solve.
The is in Quadrants III and IV, so and .
Points to Consider
 What are the angles that have and as reference angles?
 Are the only angles that we can find the exact sine, cosine, or tangent values for, multiples of ? (Recall that would be , making it a multiple of )
Review Questions
 Find the exact value for:
 If , is in quad II, and , is in quad I find
 If , is in quad III, and , is in quad II find
 Simplify:
 Prove the identity:
 Simplify
 Verify the identity:
 Simplify
 Verify that , using the sine sum formula.
 Reduce the following to a single term: .
 Prove
 Find all solutions to , when is between .
 Solve for all values of between for .
 Find all solutions to , when is between .
Review Answers

 If and in Quadrant II, then by the Pythagorean Theorem . And, if and in Quadrant I, then by the Pythagorean Theorem . So, to find and
 If and in Quadrant III, then cosine is also negative. By the Pythagorean Theorem, the second leg is , so . If the and in Quadrant II, then the cosine is also negative. By the Pythagorean Theorem, the second leg is , so . To find , plug this information into the sine sum formula.
 This is the cosine difference formula, so:
 This is the expanded sine sum formula, so:
 Step 1: Expand using the cosine sum formula and change everything into sine and cosine Step 2: Find a common denominator for the right hand side. The two sides are the same, thus they are equal to each other and the identity is true.
 Step 1: Expand and using the sine sum and difference formulas. Step 2: FOIL and simplify. Step 3: Substitute for and for , distribute and simplify.
 This could also be verified by using
 Step 1: Expand using the cosine and sine sum formulas. Step 2: Distribute and and simplify.
 Step 1: Expand left hand side using the sum and difference formulas Step 2: Divide each term on the left side by and simplify
 To find all the solutions, between , we need to expand using the sum formula and isolate the . This is true when , or
 First, solve for . Now, use the tangent sum formula to expand for when . This is true when or . If the tangent sum formula to expand for when , we get no solution as shown. Therefore, the tangent sum formula cannot be used in this case. However, since we know that when or , we can solve for as follows. Therefore, all of the solutions are
 To solve, expand each side: Set the two sides equal to each other: As a decimal, this is , so and .