#### Answer

$$(1-\cos^2\alpha)(1+\cos^2\alpha)=2\sin^2\alpha-\sin^4\alpha$$
By dealing with the left side. we prove that both sides are equal and this is thus an identity.

#### Work Step by Step

$$(1-\cos^2\alpha)(1+\cos^2\alpha)=2\sin^2\alpha-\sin^4\alpha$$
We would try with the left side first.
$$A=(1-\cos^2\alpha)(1+\cos^2\alpha)$$
We notice that $\sin^2\alpha=1-\cos^2\alpha$. That means,
$$A=\sin^2\alpha(1+\cos^2\alpha)$$
$$A=\sin^2\alpha+\sin^2\alpha\cos^2\alpha$$
Also, as we witness that the right side only includes $\sin\alpha$, it is better to change $\cos^2\alpha$ into $1-\sin^2\alpha$.
$$A=\sin^2\alpha+\sin^2\alpha(1-\sin^2\alpha)$$
$$A=\sin^2\alpha+\sin^2\alpha-\sin^4\alpha$$
$$A=2\sin^2\alpha-\sin^4\alpha$$
Thus, the left side is equal to the right side. The expression is therefore an identity.