Sum and Difference Identities Math Example 2

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Example 2

medium
Simplify sin⁡(x+y)+sin⁡(x−y)\sin(x + y) + \sin(x - y).

Solution

  1. 1
    Expand using sum and difference formulas: sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin(x+y) = \sin x \cos y + \cos x \sin y.
  2. 2
    sin⁡(x−y)=sin⁡xcos⁡y−cos⁡xsin⁡y\sin(x-y) = \sin x \cos y - \cos x \sin y.
  3. 3
    Add them: sin⁡(x+y)+sin⁡(x−y)=2sin⁡xcos⁡y\sin(x+y) + \sin(x-y) = 2\sin x \cos y.

Answer

2sin⁡(x)cos⁡(y)2\sin(x)\cos(y)
When you add the sine of a sum and the sine of a difference, the cos⁡xsin⁡y\cos x \sin y terms cancel and the sin⁡xcos⁡y\sin x \cos y terms double. This is actually the product-to-sum identity in reverse: 2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A \cos B = \sin(A+B) + \sin(A-B).

About Sum and Difference Identities

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.

Learn more about Sum and Difference Identities →

More Sum and Difference Identities Examples

Example 1 easy

Find the exact value of [formula] using the sum identity.

Example 3 medium

Find the exact value of [formula] using a difference identity.

Example 4 hard

If [formula] with [formula] in QI and [formula] with [formula] in QII, find [formula].

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