Answer
$\sin^4\theta-\cos^4\theta=2\sin^2\theta-1$
Work Step by Step
Start with the left side:
$\sin^4\theta-\cos^4\theta$
Write it as a difference of perfect squares and factor:
$=(\sin^2\theta)^2-(\cos^2\theta)^2$
$=(\sin^2\theta+\cos^2\theta)(\sin^2\theta-\cos^2\theta)$
Simplify:
$=1*(\sin^2\theta-\cos^2\theta)$
$=\sin^2\theta-\cos^2\theta$
$=\sin^2\theta-(1-\sin^2\theta)$
$=\sin^2\theta-1+\sin^2\theta$
$=2\sin^2\theta-1$
Since this equals the right side, the identity has been proven.
