Verify the identity: (1-cosx)(1+cosx) = sin^{2}x.

Question:

Verify the identity: {eq}(1-cosx)(1+cosx) = sin^{2}x. {/eq}

Proving

Verifying the identity means that we have to prove the two sides of the equation to be the same. For this, we need to simplify the left-hand side of the equation as it is more complicated and make it the same as the other side.

Answer and Explanation:

The identity can be verified as follows.

$$\begin{align} (1-\cos x)(1+\cos x) &= \sin^{2}x\\ 1-\cos^2&=\sin^2x&&&&\left [ \because (a+b)(a-b)=a^2-b^2 \right ]\\ \therefore \sin^2 x&=\sin ^2x&&&&\left [ \because \sin^2 +\cos^2x=1 \right ]\\ \end{align} $$