sin4xcos4x12cos2x verify the identitySolutionLHS sin4x cos

sin⁴x-cos⁴x=1-2cos²x verify the identity

Solution

LHS = sin^4x - cos^4x = ( sin^2x + cos^2x ) * ( sin^2x - cos^2x)

= 1 * ( sin^2x - cos^2x )
= sin^2x - cos^2x
= 1 - cos^2x - cos^2x
= 1 - 2cos^2x = RHS

= -1 ( cos^2x - sin^2x)
= -1 ( cos2x)
= -1 ( 2 cos^2x -1)
= - 2 cos ^2x + 1
= 1 - 2cos^2x = RHS