Question

If the solution set of inequality $\left(\text{cosec}^{-1} x\right)^{2}-2$ $\text{cosec}^{-1} x \geq \frac{\pi}{6}\left(\text{cosec}^{-1} x-2\right)$ is $(-\infty, m] \cup[n, \infty)$ then $(m+n)$ equals

Answer 1

$\lambda^{2}-2 \lambda \geq \frac{\pi}{6}(\lambda-2)$, where $\lambda=\text{cosec}^{-1} x$
$(\lambda-2) \cdot\left(\lambda-\frac{\pi}{6}\right) \geq 0$
$\therefore x \in(-\infty,-1] \cup[2, \infty)$
$\Rightarrow m=-1, n=2$
So, $(m+n)=1$