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)\sin(A \pm B), cos⁡(A±B)\cos(A \pm B), and tan⁡(A±B)\tan(A \pm B) in terms of sin⁡A\sin A, cos⁡A\cos A, sin⁡B\sin B, and cos⁡B\cos B.

What happens when you combine two rotations? If you rotate by angle AA and then by angle BB, 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\sin x + \cos x in the form Rsin⁡(x+ϕ)R\sin(x + \phi) with R>0R > 0, ϕ∈(−π/2,π/2)\phi \in (-\pi/2, \pi/2).

Example 2

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

Example 3

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

Example 4

hard
If tan⁡A=13\tan A = \tfrac{1}{3} and tan⁡B=12\tan B = \tfrac{1}{2}, find A+BA + B (in radians, with A,BA, B in QI).

Example 5

challenge
Given sin⁡A=35\sin A = \tfrac{3}{5} in QII and cos⁡B=−1213\cos B = -\tfrac{12}{13} in QII, find tan⁡(A+B)\tan(A + B).

Example 6

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

Example 7

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

Example 8

easy
State the formula for sin⁡(A−B)\sin(A - B).

Example 9

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

Example 10

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

Example 11

easy
State the formula for cos⁡(A−B)\cos(A - B).

Example 12

medium
Simplify sin⁡(A+B)+sin⁡(A−B)cos⁡(A+B)+cos⁡(A−B)\dfrac{\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)]\sin A\cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)].

Example 14

medium
If sin⁡A=35\sin A = \tfrac{3}{5} in QI and sin⁡B=1213\sin B = \tfrac{12}{13} in QI, find sin⁡(A−B)\sin(A - B).

Example 15

easy
State the formula for cos⁡(A+B)\cos(A + B).

Example 16

medium
If cos⁡A=−35\cos A = -\tfrac{3}{5} in QII and sin⁡B=−513\sin B = -\tfrac{5}{13} in QIII, find cos⁡(A−B)\cos(A - B).

Example 17

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

Example 18

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

Example 19

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

Example 20

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