\(D:\theta
eq\frac{k\pi}{2};\ k\in\mathbb{Z}\\\\tan\theta(csc\theta+2)=0\iff tan\theta=0\ \vee\ csc\theta+2=0\\\\\\tan\theta=0\iff\theta=k\pi
otin D\\\\csc\theta+2=0\\\\csc\theta=-2\\\\\frac{1}{sin\theta}=-2\\\\-2sin\theta=1\ /:(-2)\\\\sin\theta=-\frac{1}{2}\)
\((\theta=-\frac{\pi}{6}+2k\pi\ \vee\ \theta=\frac{7\pi}{6}+2k\pi)-answer\)