Question

Prove the identity: cos 4x = 1 – 8 cos² x + 8 cos⁴ x

Answer 1

Let us consider the LHS

cos 4x

As we know, cos 2x = 2 cos² x – 1

Therefore,

cos 4x = 2 cos² 2x – 1

= 2(2 cos² 2x – 1)² – 1

= 2[(2 cos² 2x)² + 1² – 2 × 2 cos² x] – 1

= 2(4 cos⁴ 2x + 1 – 4 cos² x) – 1

= 8 cos⁴ 2x + 2 – 8 cos² x – 1

= 8 cos⁴ 2x + 1 – 8 cos² x

= RHS

Thus proved.