Prove that

To Prove:

Taking LHS,

= cos⁴x + sin⁴x

Adding and subtracting 2sin²x cos²x, we get

= cos⁴x + sin⁴x + 2sin²x cos²x – 2sin²x cos²x

We know that,

a² + b² + 2ab = (a + b)²

= (cos²x + sin²x) – 2sin²x cos²x

= (1) – 2sin²x cos²x [∵ cos² θ + sin² θ = 1]

= 1 – 2sin²x cos²x

Multiply and divide by 2, we get

[∵ sin 2x = 2 sinx cosx]

= RHS

∴ LHS = RHS

Hence Proved