Express each of the following as an algebraic sum of sines or cosines :

Solution:

(i) $2 \sin 6 x \cos 4 x=\sin (6 x+4 x)+\sin (6 x-4 x)$

$=\sin 10 x+\sin 2 x$

Using,

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

(ii) $2 \cos 5 x \sin 3 x=\sin (5 x+3 x)-\sin (5 x-3 x)$

$=\sin 8 x-\sin 2 x$

Using,

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

(iii) $2 \cos 7 x \cos 3 x=\cos (7 x+3 x)+\cos (7 x-3 x)$

$=\cos 10 x+\cos 4 x$

Using,

$2 \cos A \cos B=\cos (A+B)+\cos (A-B)$

(iv) $2 \sin 8 x \sin 2 x=\cos (8 x-2 x)-\cos (8 x+2 x)$

$=\cos 6 x-\cos 10 x$

Using,

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