Question

Prove the following:

cos²2x − cos²6x = sin4x sin8x

Answer 1

L.H.S. = cos²2x − cos²6x 

= (cos 2x)² − (cos 6x)²

= (cos 2x + cos 6x) (cos 2x − cos 6x)

= `[2cos((2x + 6x)/2) cos((2x - 6x)/2)]*[2sin((2x + 6x)/2)sin((6x - 2x)/2)]`

= [2 cos 4x cos (− 2x)] [2 sin 4x sin 2x]

= (2 cos 4x cos 2x) (2 sin 4x sin 2x)

= (2 sin 2x cos 2x) (2 sin 4x cos 4x)

= sin 4x sin 8x

= R.H.S.