Question

Sin^4 x + cos^4 x = 1-2 sin^2 x cos^2 x

Answer 1

To prove that sin^4 x + cos^4 x = 1 - 2 sin^2 x cos^2 x

LHS = sin^4 x + cos^4 x

Add and subtract 2 sin^2 x cos^2 x to the LHS

= [sin^4 x + 2 sin^2 x cos^2 x + cos^4 x] - 2 sin^2 x cos^2 x

= [sin^2 x + cos^2 x]^2 - 2 sin^2 x cos^2 x

= 1^2 - 2 sin^2 x cos^2 x

= 1 - 2 sin^2 x cos^2 x = RHS. Proved.