Practice Sum and Difference Identities in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Formulas that express sin⁡(A±B), cos⁡(A±B), and tan⁡(A±B) in terms of sin⁡A, cos⁡A, sin⁡B, and cos⁡B.

What happens when you combine two rotations? If you rotate by angle A and then by angle B, the result involves both angles interacting. The sum and difference formulas tell you exactly how the trig values of two separate angles combine. They're like a multiplication rule for rotations—the result isn't simply adding the trig values, but mixing sines and cosines together.

Showing a random 20 of 50 problems.

Example 1

hard
Express sin⁡x+cos⁡x in the form Rsin⁡(x+ϕ) with R>0, ϕ∈(−π/2,π/2).

Example 2

medium
Simplify sin⁡50°cos⁡20°−cos⁡50°sin⁡20°.

Example 3

easy
Compute sin⁡30°cos⁡60°+cos⁡30°sin⁡60°.

Example 4

hard
If tan⁡A=13 and tan⁡B=12, find A+B (in radians, with A,B in QI).

Example 5

challenge
Given sin⁡A=35 in QII and cos⁡B=−1213 in QII, find tan⁡(A+B).

Example 6

challenge
Show that cos⁡(A+B)cos⁡(A−B)=cos⁡2A−sin⁡2B.

Example 7

medium
Compute cos⁡15° exactly using 15°=45°−30°.

Example 8

easy
State the formula for sin⁡(A−B).

Example 9

medium
Compute sin⁡75° exactly using 75°=45°+30°.

Example 10

easy
Fill in: sin⁡(A−B)=sin⁡Acos⁡B− ___.

Example 11

easy
State the formula for cos⁡(A−B).

Example 12

medium
Simplify sin⁡(A+B)+sin⁡(A−B)cos⁡(A+B)+cos⁡(A−B).

Example 13

hard
Prove the product-to-sum formula sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)].

Example 14

medium
If sin⁡A=35 in QI and sin⁡B=1213 in QI, find sin⁡(A−B).

Example 15

easy
State the formula for cos⁡(A+B).

Example 16

medium
If cos⁡A=−35 in QII and sin⁡B=−513 in QIII, find cos⁡(A−B).

Example 17

challenge
Prove that sin⁡(A+B)+sin⁡(A−B)=2sin⁡Acos⁡B.

Example 18

easy
Express cos⁡(π−x) using the difference formula.

Example 19

medium
Compute tan⁡75° exactly using 75°=45°+30°.

Example 20

easy
Express sin⁡(90°−θ) using the difference formula.
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