Verify the identity: cos 4x + cos 2x = 2 - 2sin²(2x) - 2sin²(x)

Solution:

We have to verify the given identity.

LHS: cos(4x) + cos(2x)

By using trigonometric identity,

cos(A + B) = cosAcosB - sinAsinB

cos(4x) = cos(2x + 2x)

cos(2x + 2x) = cos(2x)cos(2x) - sin(2x)sin(2x)

= cos²(2x) - sin²(2x)

By using trigonometric identity,

cos²x + sin²x = 1

cos²x = 1 - sin²x

So, cos²(2x) - sin²(2x) = 1 - sin²(2x) - sin²(2x)

cos(4x) = 1 - 2sin²(2x)

By using trigonometric identity,

cos(2A) = 1 - 2sin²A

So, cos(2x) = 1 - 2sin²x

Now, cos(4x) + cos(2x) = 1 - 2sin²(2x) + 1 - 2sin²(x)

= 2 - 2sin²(2x) - 2sin²(x)

= RHS

LHS = RHS

Alternative solution:

It is given that

cos 4x + cos 2x = 2 - 2 sin² 2x - 2 sin2 x

Consider the LHS

We know that cos 2x = 1 - 2 sin² x [using the cos 2x formula]

LHS = cos 4x + cos 2x

Here cos 4x = 1 - 2 sin² 2x

cos 2x = 1 - 2 sin² x

LHS = 1 - 2sin² 2x + 1 - 2 sin² x

LHS = 2 - 2sin² 2x - 2 sin² x

=RHS

Therefore, the identity is verified.

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Verify the identity: cos 4x + cos 2x = 2 - 2sin²(2x) - 2sin²(x)

Summary:

The identity cos 4x + cos 2x = 2 - 2sin²(2x) - 2sin²(x) is verified.