To prove: cos 4x = 1 – 8 cos² x + 8 cos⁴ x

Proof:

Take LHS:

cos 4x

Identities used:

cos 2x = = 2 cos² x – 1

Therefore,

= 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

Hence Proved