Question

Rewrite sin (6x) sin (x) as a sum or difference.

Answer 1

Recall that

`\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y`

It follows that

`\sin x\sin y=\frac{\cos(x-y)-\cos(x+y)}2`

Replace
`y` with
`6x` and you get

`\sin6x\sin x=\frac{\cos(x-6x)-\cos(x+6x)}2=\frac{\cos5x-\cos7x}2`

(using the fact that
`\cos(-5x)=\cos5x`)