Question

Prove:

sin^2x - sin^4x = cos^2x - cos^4x




What I have,


LS
= (sinx - sin^2x) (sinx + sin^2x)
= (sinx - 1 -cos^2x) (sinx + 1 - cos^2x)
= sin^2x + sinx - sinx - cos^2xsinx - cos^2xsinx - 1 - 1 + cos^4x
= sin^2x - 2cos^2xsinx - 2 + cos^4x



Where did I go wrong? Can anyone please help me prove this identity?

Answer 1

(sinx - 1 -cos^2x) (sinx + 1 - cos^2x)

should have been

(sinx - 1 <b>+</b> cos^2x) (sinx + 1 - cos^2x) and then the next line should be

sin^2x + sinx - cos^2xsinx - sinx - 1 + cos^2 x + cos^2xsinx - cos^4x
= cos^2 x - cos^4 x
= RS

an easier way would have been

LS = sin^2x(1-sin^2x) by common factor
=sin^2x(cos^2x)

RS = cos^2x(1-cos^2x) also by common factor
=cos^2x(sin^2x)
= LS