Find the general solution of cosec x = –2
Question:

Find the general solution of $\operatorname{cosec} x=-2$

Solution:

$\operatorname{cosec} x=-2$

It is known that

$\operatorname{cosec} \frac{\pi}{6}=2$

$\therefore \operatorname{cosec}\left(\pi+\frac{\pi}{6}\right)=-\operatorname{cosec} \frac{\pi}{6}=-2$ and $\operatorname{cosec}\left(2 \pi-\frac{\pi}{6}\right)=-\operatorname{cosec} \frac{\pi}{6}=-2$

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

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

Now, $\operatorname{cosec} x=\operatorname{cosec} \frac{7 \pi}{6}$

$\Rightarrow \sin x=\sin \frac{7 \pi}{6} \quad\left[\operatorname{cosec} x=\frac{1}{\sin x}\right]$

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

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