Prove that:cos2x × cos(x/2) − cos3x × cos × (9x/2) = sin5x × sin × (5x/2)

Question

Shiksha Nation · Accepted Answer

L.H.S. = 1/2 [2cos2x × cos(x/2) − 2cos(9x/2) cos(3x)]

= 1/2 [cos(2x + x/2) + cos(2x − x/2) − cos(9x/2 + 3x) − cos(9x/2 − 3x)]

= 1/2 [cos(5x/2) + cos(3x/2) − cos(15x/2) − cos(3x/2)] = 1/2 [cos(5x/2) − cos(15x/2)]

= 1/2 [−2sin((5x + 15x)/2) × sin((5x − 15x)/2)]

= −sin(5x) × sin(−5x/2) = sin(5x) × sin(5x/2) = R.H.S.

Prove that:
cos2x × cos(x/2) − cos3x × cos × (9x/2) = sin5x × sin × (5x/2)