Question 109301
I'm going to use the double angle identities *[Tex \Large \sin(2\theta)=2\sin(\theta)\cos(\theta)], *[Tex \Large \cos(2\theta)=\cos^{2}(\theta)-\sin^{2}(\theta)], and *[Tex \Large \tan(2\theta)=\frac{2\tan(\theta)}{1-\tan(\theta)}]


Notice how we have *[Tex \LARGE \cos(\theta)\sin(\theta)] in the problem. So we must divide both sides of the identity *[Tex \Large \sin(2\theta)=2\sin(\theta)\cos(\theta)] by 2 to get *[Tex \Large \frac{1}{2}\sin(2\theta)=\sin(\theta)\cos(\theta)]









*[Tex \LARGE \begin{align*} \frac{\cos(\theta)\sin(\theta)}{\cos^{2}(\theta)-\sin^{2}(\theta)}&=\frac{\tan(\theta)}{1-\tan(\theta)} \\ \frac{\frac{1}{2}\sin(2\theta)}{\cos^{2}(\theta)-\sin^{2}(\theta)}&= \mbox{ Use the first identity listed}\\ \frac{\frac{1}{2}\sin(2\theta)}{\cos(2\theta)}&=\mbox{ Use the second identity listed}\\ \frac{1}{2}\tan(2\theta)&= \mbox{ Replace sine over cosine with tangent}\\ \frac{1}{2}\frac{2\tan(\theta)}{1-\tan(\theta)}&= \mbox{ Use the third identity listed}\\ \frac{\tan(\theta)}{1-\tan(\theta)}&= \mbox{ Simplify}\\ \end{align*}]