Answer
All solutions: $\frac{\pi}{6}+2k\pi$, $\frac{5\pi}{6}+2k\pi$, $\frac{3\pi}{4}+k\pi$
Solutions in $[0, 2\pi)$: $\frac{\pi}{6}$, $\frac{3\pi}{4}$, $\frac{5\pi}{6}$, $\frac{7\pi}{4}$
Work Step by Step
$2\sin\theta\tan\theta-\tan\theta=1-2\sin\theta$
$2\sin\theta\tan\theta-\tan\theta+2\sin\theta-1=0$
$\tan\theta(2\sin\theta-1)+2\sin\theta-1=0$
$(\tan\theta+1)(2\sin\theta-1)=0$
If $\tan\theta+1=0$, then $\tan\theta=-1$, and the solution is $\frac{3\pi}{4}+k\pi$. (The solutions in $[0, 2\pi)$ are $\frac{3\pi}{4}$ and $\frac{7\pi}{4}$.)
If $2\sin\theta-1=0$, then $2\sin\theta=1$, and $\sin \theta=\frac{1}{2}$. The solutions are $\frac{\pi}{6}+2k\pi$ and $\frac{5\pi}{6}+2k\pi$. (The solutions in $[0, 2\pi)$ are $\frac{\pi}{6}$ and $\frac{5\pi}{6}$.)
