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

$\sin x\sin y=\dfrac{\cos(x-y)-\cos(x+y)}2$

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

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

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