Prove that $\cot 2\theta + \tan 2\theta = 2\sin 4\theta$

Full solution

Q. Prove that

\cot 2\theta + \tan 2\theta = 2\sin 4\theta

Rewrite $\cot 2\theta$ : Rewrite $\cot 2\theta$ in terms of $\tan 2\theta$ . $\newline$ $\cot 2\theta = \frac{1}{\tan 2\theta}$
Add $\cot^2\theta$ and $\tan^2\theta$ : Add $\cot^2\theta$ and $\tan^2\theta$ . $\newline$ $\cot^2\theta + \tan^2\theta = \frac{1}{\tan^2\theta} + \tan^2\theta$
Find common denominator: Find a common denominator and combine the terms. $\newline$ $(1 + \tan^2(2\theta)) / \tan(2\theta) = 2\sin(4\theta)$
Use Pythagorean identity: Use the Pythagorean identity: $1 + \tan^2(2\theta) = \sec^2(2\theta)$ .
$\frac{\sec^2(2\theta)}{\tan 2\theta} = 2\sin 4\theta$
Convert $\sec^2(2\theta)$ : Convert $\sec^2(2\theta)$ to $\frac{1}{\cos^2(2\theta)}$ and simplify. $\frac{\frac{1}{\cos^2(2\theta)}}{\frac{\sin 2\theta}{\cos 2\theta}} = 2\sin 4\theta$
Multiply by $\cos^2(2\theta)$ : Multiply both sides by $\cos^2(2\theta)$ to clear the fraction. $\newline$ $\sin 2\theta = 2\sin 4\theta \cdot \cos^2(2\theta)$
Use double angle identity: Use the double angle identity for sine: $\sin 4\theta = 2\sin 2\theta \cdot \cos 2\theta$ . $\sin 2\theta = 2 \cdot 2\sin 2\theta \cdot \cos 2\theta \cdot \cos^2(2\theta)$
Divide by $\sin 2\theta$ : Divide both sides by $\sin 2\theta$ , assuming $\sin 2\theta$ is not zero. $\newline$ $1 = 4 \cdot \cos 2\theta \cdot \cos^2(2\theta)$
Use Pythagorean identity: Use the Pythagorean identity: $\cos^2(2\theta) = 1 - \sin^2(2\theta)$ . $\newline$ $1 = 4 \cdot \cos^2(\theta) \cdot (1 - \sin^2(2\theta))$