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I'm having massive issues proving this identity:

$$\cot^2 (2x) + \cos^2 (2x) + \sin^2 (2x) = \csc^2 (2x)$$

How is this proven?

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Recall the following identities $$\cos^2{\alpha} + \sin^2{\alpha} = 1$$ $$\cot^2{\beta} + 1 = \csc^2 \beta$$ Hence, $$\cot^2 (2x) + \cos^2 (2x) + \sin^2 (2x) = \underbrace{\cot^2 (2x) + \underbrace{\left(\cos^2 (2x) + \sin^2 (2x) \right)}_{=1}}_{\cot^2(2x)+1 = \csc^2(2x)}$$

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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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