Need help Proving Identities Prove the Identity: $$\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta + 1}{\sin \theta \cos \theta}.  $$

Since Stonebrakermatt has started from the LHS to prove this identity, I will start from the RHS. Using the well-known identity $\cos^2x+\sin^2x=1$, then
\begin{align}
\require{cancel}
\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} &= \frac{\cos \theta + 1}{\sin \theta \cos \theta}\\
&=\frac{\cos \theta + \cos^2\theta+\sin^2\theta}{\sin \theta \cos \theta}\\
&=\frac{\cos \theta + \cos^2\theta}{\sin \theta \cos \theta}+\frac{\sin^2\theta}{\sin \theta \cos \theta}\\
&=\frac{\cancel{\cos \theta} (1+ \cos\theta)}{\sin \theta \cancel{\cos \theta}}+\frac{\cancel{\sin \theta}\sin\theta}{\cancel{\sin \theta} \cos \theta}\\
&=\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta}&\blacksquare
\end{align}
