Find the general solution of $\cos ec\, x=-2$

$\cos ec\, x=-2$

It is known that

$\cos ec\, \frac{\pi}{6}=2$

$\therefore \cos ec \left(\pi+\frac{\pi}{6}\right)=-\cos ec\, \frac{\pi}{6}=-2$ and $\cos ec\, \left(2 \pi-\frac{\pi}{6}\right)=-\cos ec\, \frac{\pi}{6}=-2$

i.e., $\cos ec\, \frac{7 \pi}{6}=-2$ and $\cos ec\, \frac{11 \pi}{6}=-2$

Therefore, the principal solutions are $x=\frac{7 \pi}{6}$ and $\frac{11 \pi}{6}$

Now $\cos ec\,  x=\cos ec\, \frac{7 \pi}{6}$

$\Rightarrow \sin x=\sin \frac{7 \pi}{6} \quad\left[\cos ec\, x=\frac{1}{\sin x}\right]$

$\Rightarrow x=n \pi+(-1)^{n} \frac{7 \pi}{6},$ where $n \in Z$

Therefore, the general solution is $x=n \pi+(-1)^{n} \frac{7 \pi}{6},$ where $n \in Z$.