# How do you prove cot2x = (1+cos4x) / (sin4x)?

Jan 20, 2016

We will use the following identities:

• $\sin \left(2 \theta\right) = 2 \sin \left(\theta\right) \cos \left(\theta\right)$

• $\cos \left(2 \theta\right) = 2 {\cos}^{2} \left(\theta\right) - 1$

Now, starting from the right hand side:

$\frac{1 + \cos \left(4 x\right)}{\sin} \left(4 x\right) = \frac{1 + \left(2 {\cos}^{2} \left(2 x\right) - 1\right)}{2 \sin \left(2 x\right) \cos \left(2 x\right)}$

$= \frac{2 {\cos}^{2} \left(2 x\right)}{2 \sin \left(2 x\right) \cos \left(2 x\right)}$

$= \cos \frac{2 x}{\sin} \left(2 x\right)$

$= \cot \left(2 x\right)$