Originally Posted by
TheEmptySet Multiply the numerator and denominator by $\displaystyle \cos^2(x)$
$\displaystyle \frac{\cos^2(x)}{\cos^2(x)}\left(\frac{1-\tan^2(x)}{1+\tan^2(x)} \right)=\frac{\cos^2(x)-\sin^2(x)}{\cos^2(x)+\sin^2(x)}=\cos^2(x)-\sin^2(x)$