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

I will assume you meant to write the following:
$$
\frac{1+ \cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} = \frac{\cos \theta + 1}{\sin \theta \cos \theta}
$$
Therefore, you can combine the fractions on the left side to get the following:
$$
\frac{\cos \theta + \cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} = \frac{\cos \theta + 1}{\sin \theta \cos \theta}
$$
Then, using the identity $\cos^2 \theta + \sin^2 \theta = 1$, we get:
$$
\frac{\cos \theta + 1}{\sin \theta \cos \theta} = \frac{\cos \theta + 1}{\sin \theta \cos \theta}
$$
Therefore, this is true.
