Answer
$$\cot4\theta=\frac{1-\tan^22\theta}{2\tan2\theta}$$
2 sides are equal as shown in the work step by step, and the equation is an identity.
Work Step by Step
$$\cot4\theta=\frac{1-\tan^22\theta}{2\tan2\theta}$$
This time we do a little bit differently. We take from the left side.
$$X=\cot4\theta$$
As in Reciprocal Identities: $\cot\theta=\frac{1}{\tan\theta}$, it means $\cot4\theta=\frac{1}{\tan4\theta}$
$$X=\frac{1}{\tan4\theta}$$
$$X=\frac{1}{\tan(2\times2\theta)}$$
For $\tan(2\times2\theta)$, we apply Double-Angle Identity for $\tan2A$, which states
$$\tan2A=\frac{2\tan A}{1-\tan^2A}$$
With $A=2\theta$, we have
$$\tan(2\times2\theta)=\frac{2\tan2\theta}{1-\tan^22\theta}$$
Thus, $$X=\frac{1}{\frac{2\tan2\theta}{1-\tan^22\theta}}$$
$$X=\frac{1-\tan^22\theta}{2\tan2\theta}$$
Therefore, $$\cot4\theta=\frac{1-\tan^22\theta}{2\tan2\theta}$$
2 sides are thus equal, and the equation is an identity.
