Question

Use the sum-to-product identities to rewrite the following expression as a product.sin(x) – sin(5x)

Answer 1

Using the sum-to-product identity:

$\sin A-\sin B=2\cos (\frac{A+B}{2})\sin (\frac{A-B}{2})$

we can replace our expression and solve, where A = x and B = 5x:

$\sin (x)-\sin (5x)=2\cos (\frac{x+5x}{2})\sin (\frac{x-5x}{2})$

Simplifying:

$\sin (x)-\sin (5x)=2\cos (\frac{6x}{2})\sin (\frac{-4x}{2})$$\sin (x)-\sin (5x)=2\cos (3x)\sin (-2x)$

Answer:

$=2\cos (3x)\sin (-2x)$