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 ((A+B)/(2))\sin ((A-B)/(2))`

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

`\sin (x)-\sin (5x)=2\cos ((x+5x)/(2))\sin ((x-5x)/(2))`

Simplifying:

`\sin (x)-\sin (5x)=2\cos ((6x)/(2))\sin ((-4x)/(2))`
`\sin (x)-\sin (5x)=2\cos (3x)\sin (-2x)`

Answer:

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