<s>[INST] Prove $\cot^2 (2x) + \cos^2 (2x) + \sin^2 (2x) = \csc^2 (2x)$ I'm having massive issues proving this identity:
$$\cot^2 (2x) + \cos^2 (2x) + \sin^2 (2x) = \csc^2 (2x)$$
How is this proven?
 [/INST] We know that $\cos^2(\theta) + \sin^2(\theta) = 1$, for any $\theta$ including $\theta = 2x$
Hence the left hand side of the equation is $\cot^2(2x) + 1$
Look at our first identity, let's divide it by $\sin^2(\theta)$ yielding
$\displaystyle (\cos^2(\theta))/(\sin^2(\theta)) + 1 = \frac{1}{\sin^2(\theta)}$
And gives us
$\displaystyle\cot^2(\theta) +1 = \frac{1}{\sin^2(\theta)}$
we let $\theta = 2x$ and we are done.
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