To Prove: cos4x + sin4x = \(\frac{1}{2}\)(2 - sin22x)
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 [∵ cos2θ + sin2θ = 1]
= 1 – 2sin2x cos2x
Multiply and divide by 2, we get
= RHS
∴ LHS = RHS
Hence Proved