1.

Prove the identity: 1 + cos2 2x = 2(cos4 x + sin4 x)

Answer»

Let us consider the LHS

1 + cos2 2x

As we know, cos2x = cos2 x – sin2 x

cos2 x + sin2 x = 1

Therefore,

1 + cos2 2x = (cos2 x + sin2 x)2 + (cos2 x – sin2 x)2

= (cos4 x + sin4 x + 2 cos2 x sin2 x) + (cos4 x + sin4 x – 2 cos2 x sin2 x)

= cos4 x + sin4 x + cos4 x + sin4 x

= 2 cos4 x + 2 sin4 x

= 2(cos4 x + sin4 x)

= RHS

Thus proved.



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