To prove cos 6x = 32 cos⁶ x – 48cos⁴ x + 18 cos² x – 1

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

LHS = cos 6x

= cos 3(2x)

= 4 cos³ 2x – 3 cos 2x [cos 3A = 4 cos3 A – 3 cos A]

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

= 4 [(2 cos² x)³ – (1)³ – 3 (2 cos² x)² + 3 (2 cos² x)] – 6cos² x + 3

= 4 [8cos⁶ x – 1 – 12 cos⁴ x + 6 cos² x] – 6 cos² x + 3

= 32 cos⁶ x – 4 – 48 cos⁴ x + 24 cos² x – 6 cos² x + 3

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

∴ LHS = RHS

Hence, Proved.