Factor a difference of squares:
[tex]\sin^4x-\cos^4x=(\sin^2x-\cos^2x)(\sin^2x+\cos^2x)[/tex]
This reduces to
[tex]\sin^2x-\cos^2x[/tex]
due to the Pythagorean identity. By the same identity, you have
[tex]\sin^2x-\cos^2x=\sin^2x-(1-\sin^2x)=2\sin^2x-1[/tex]
and you're done.
