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

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Solve complex trigonomentric equations

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Q. Prove that

\cot 2\theta + \tan 2\theta = \frac{2}{\sin 4\theta}

Rewrite $\cot 2\theta$ : Rewrite $\cot 2\theta$ as $\frac{1}{\tan 2\theta}$ to have a common base for addition. $\newline$ $\cot 2\theta + \tan 2\theta = \frac{1}{\tan 2\theta} + \tan 2\theta$
Combine terms over denominator: Combine the terms over a common denominator. $\newline$ $(1 + \tan^2(2\theta)) / \tan(2\theta) = \frac{2}{\sin(4\theta)}$
Use Pythagorean identity: Use the Pythagorean identity: $1 + \tan^2(2\theta) = \sec^2(2\theta)$ .
$\frac{\sec^2(2\theta)}{\tan 2\theta} = \frac{2}{\sin 4\theta}$
Express $\sec^2(2\theta)$ : Express $\sec^2(2\theta)$ as $\frac{1}{\cos^2(2\theta)}$ and simplify. $\frac{\frac{1}{\cos^2(2\theta)}}{\frac{\sin 2\theta}{\cos 2\theta}} = \frac{2}{\sin 4\theta}$
Simplify by multiplying: Simplify the left side by multiplying by $\frac{\cos 2\theta}{\sin 2\theta}$ . $\frac{1}{\sin 2\theta} = \frac{2}{\sin 4\theta}$
Use double angle identity: Use the double angle identity for sine: $\sin 4\theta = 2\sin 2\theta \cos 2\theta$ . $\frac{1}{\sin 2\theta} = \frac{2}{2\sin 2\theta \cos 2\theta}$
Cancel out common terms: Cancel out the common terms. $\newline$ $1 = \frac{1}{\cos 2\theta}$
Take reciprocal to solve: Take the reciprocal of both sides to solve for $\cos 2\theta$ . $\cos 2\theta = 1$
Find values of $2\theta$ : Find the values of $2\theta$ that satisfy the equation $\cos 2\theta = 1$ . $\newline$ $2\theta = 2n\pi$ , where $n$ is an integer.
Divide to solve for theta: Divide by $2$ to solve for theta. $\newline$ $\theta = n\pi$ , where $n$ is an integer.

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