<s>[INST] Prove $\frac{\sin\theta}{1+\cos\theta} + \frac{1+\cos\theta}{\sin\theta} = \frac{2}{\sin\theta}$ How to prove:
$\frac{\sin\theta}{1+\cos\theta} + \frac{1+\cos\theta}{\sin\theta} = \frac{2}{\sin\theta}$
Please help.
 [/INST] Let $u=\theta/2$. Then, by using double-angle formulas, $\sin \theta = 2 \cos u \sin u$, $1+\cos \theta = 2\cos^2 u$.
So the left-hand side is equal to
$$
\frac{2\cos u \sin u}{2\cos^2 u} + \frac{2 \cos^2 u}{2\cos u \sin u} = \frac{\cos u}{\sin u} + \frac{\sin u}{\cos u} = \frac{\cos^2 u + \sin^2 u}{\cos u \sin u}=\frac{1}{\cos u \sin u} \, ,
$$
while the right-hand side is equal to
$$
\frac{2}{2 \cos u \sin u}=\frac{1}{\cos u \sin u} \, .
$$
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