Need help Proving Identities

Question

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

Answer 1

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.

Answer 2

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}