Question

Prove that cos 6x = 32 cos⁶ x- 48 cos⁴ x + 18 cos² x- 1.

Answer 1

L.H.S. = cos 6x = cos 3(2x) [∵ cos 3x = 4 cos³ x – 3 cos x]

= 4 cos³ 2x – 3 cos 2x

= cos 2x (4 cos² 2x – 3)

= cos 2x [4(cos 2x)² – 3] [∵ cos 2x = 2 cos² x – 1]

= cos 2x [4(2 cos² x – l)² – 3]

= cos 2x [4(4 cos⁴ x + 1 – 4 cos² x) – 3]

= cos 2x (16 cos⁴ x + 4 – 16 cos² x – 3)

= (2 cos² x – 1) (16 cos⁴ x + 4 – 16 cos² x – 3)

= 32 cos⁶ x + 8 cos² x – 32 cos⁴ x – 6 cos² x – 16 cos⁴ x – 4 + 16 cos² x + 3

= 32 cos⁶ x – 48 cos⁴ x + 18 cos² x – 1

= R.H.S.  Hence Proved.