Express each of the following as a product.

Solution:

1. $\sin 10 x+\sin 6 x=2 \sin \frac{10 x+6 x}{2} \cos \frac{10 x-\sin x}{2}$

$=2 \sin \frac{18 x}{2} \cos \frac{4 x}{2}$

$=2 \sin 9 x \cos 2 x$

Using,

$\sin (A+B)=\sin A \cos B+\cos A \sin B$

2. $\sin 7 x-\sin 3 x=2 \cos \frac{7 x+3 x}{2} \sin \frac{7 x-3 x}{2}$

$=2 \cos \frac{10 x}{2} \sin \frac{4 x}{2}$

$=2 \cos 5 x \sin 2 x$

Using,

$\sin (A-B)=\sin A \cos B-\cos A \sin B$

3. $\cos 7 x+\cos 5 x=2 \cos \frac{7 x+5 x}{2} \cos \frac{7 x-5 x}{2}$

$=2 \cos \frac{12 x}{2} \cos \frac{2 x}{2}$

$=2 \cos 6 x \cos x$

Using,

$\cos (A+B)=\cos A \cos B-\sin A \sin B$

4. $\cos 2 x-\cos 4 x=-2 \sin \frac{2 x+4 x}{2} \sin \frac{2 x-4 x}{2}$

$=-2 \sin \frac{6 x}{2} \sin \frac{-2 x}{2}$

$=2 \sin 3 x \sin x$

Using,

$\cos (A-B)=\cos A \cos B+\sin A \sin B$