Question

Q6 Q11.) Express the given product as a sum or difference containing only sines or cosines.

Answer 1

Need to use two expressions:

sin(A + B) = sin Acos B + cos A sin B, and

sin(A − B) = sin A cos B − cos A sin B

sin(A + B) - sin(A − B) = 2cos A sin B

cos A sin B = 1/2[sin(A + B) - sin(A − B)]

Substituting

cos 3x/2 sin x/2 = 1/2[sin(3x/2 + x/2) - sin(3x/2 − x/2)]

= 1/2[sin(2x) - sin(x)]

Answer 2

let y=x/2
cos(3x/2)sin(x/2)=cos(3y)sin(y)
use sin(x+y) n sin(x-y) eqns
2cos(3y)sin(y)=sin(3y+y)-sin(3y-y)
cos(3y)sin(y)=1/2sin(4y)-1/2sin2y
ans is 1/2sin(2x)-1/2sin(x)