Question

Prove that cos 2x cos x/2 - cos 3x cos 9x/2 = sin 5x sin 5x/2

Answer 1

Solution: We have

cos 2x cos(x/2) - cos 3x cos ( 9x/2 )

multiply and divide by 2, we get

1/2 [ 2 cos 2x cos ( x/2) - 2 cos 3x cos ( 9x/2 )]

using the formula ,

2 cos x cos y = cos(x+y) + cos(x-y) , we get ,

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

= 1/2 [ cos 5x/2 - cos 3x/2 - cos 15x/2 + cos3x/ 2]

= 1/2 [ cos 5x/2 - cos 15x/2]

= 1/2 [ -(- 2  sin 5x sin 5x/2 )]

= sin 5x sin 5x/2