1.

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

Answer»

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.



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