Answer
$\frac{\pi}{8}$, $\frac{3\pi}{8}$, $\frac{5\pi}{8}$, $\frac{7\pi}{8}$, $\frac{9\pi}{8}$, $\frac{11\pi}{8}$, $\frac{13\pi}{8}$, $\frac{15\pi}{8}$
Work Step by Step
$\tan \theta+\cot \theta=4\sin 2\theta$
$\frac{\sin \theta}{\cos\theta}+\frac{\cos \theta}{\sin \theta}=4\sin 2\theta$
Multiply both sides by $\sin \theta\cos \theta$:
$\sin^2\theta+\cos ^2\theta=4\sin 2\theta\sin\theta\cos\theta$
$1=4\sin 2\theta\sin\theta\cos\theta$
$1=2\sin 2\theta(2\sin\theta\cos\theta)$
$\frac{1}{2}=\sin 2\theta\sin 2\theta$
$\frac{1}{2}=\sin^2 2\theta$
$\sin 2\theta=\pm \frac{\sqrt{2}}{2}$
If $\sin 2\theta=\frac{\sqrt{2}}{2}$, then $2\theta=\dots$, $\frac{\pi}{4}$, $\frac{3\pi}{4}$, $\frac{9\pi}{4}$, $\frac{11\pi}{4}$, $\dots$. The only solutions for $\theta$ in $[0, 2\pi)$ are $\frac{\pi}{8}$, $\frac{3\pi}{8}$, $\frac{9\pi}{8}$, and $\frac{11\pi}{8}$.
If $\sin 2\theta=-\frac{\sqrt{2}}{2}$, then $2\theta=\dots$, $\frac{5\pi}{4}$, $\frac{7\pi}{4}$, $\frac{13\pi}{4}$, $\frac{15\pi}{4}$, $\dots$. The only solutions for $\theta$ in $[0, 2\pi)$ are $\frac{5\pi}{8}$, $\frac{7\pi}{8}$, $\frac{13\pi}{8}$, and $\frac{15\pi}{8}$.
