<s>[INST] Prove: $\csc a +\cot a = \cot\frac{a}{2}$ 
Prove: $$\csc a + \cot a = \cot\frac{a}{2}$$

All I have right now, from trig identities, is 
$$\frac{1}{\sin a} + \frac{1}{\tan a} = \frac{1}{\tan(a/2)}$$
Where do I go from there?
 [/INST] We start with the following identities: $\quad\sin(2a) = 2\sin a \cos a\quad$ $\quad\cos(2a) = 1-2\sin^2 a\quad$
We solve these to get the half-angle identities: $\quad \sin(a) = 2\sin \frac a2 \cos \frac a2\quad$$\quad\sin^2 \frac a2 = \frac 12 (1-\cos a)$
We now tackle the problem
$$\frac{1}{\sin a} + \frac{\cos a}{\sin a} = \frac{\cos \frac a2}{\sin \frac a2}$$
multiplying out both sides we get that
$$2\sin^2 \frac a2(\cos a + 1) =2\sin \frac a2\cos \frac{a}{2} \sin a$$
Using the identities above we get that
$$(1-\cos a)(1+\cos a) = \sin^2 a$$
$$\implies 1 - \cos^2 a = \sin^2 a$$
$$\implies 1  = \sin^2 a + \cos^2 a$$
We now have a trivial trigonometric identity, so the equivalence is proved
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