Question

If $\sum _{i=1}^{10}{\sin }^{-1}{x}_i=5\pi$, then the value of $\sum _{i=1}^{10}{x}_i^2$ is

Answer 1

Given : $\sum _{i=1}^{10}{\sin }^{-1}{x}_i=5\pi$
$\Rightarrow {\sin }^{-1}{x}_1+{\sin }^{-1}{x}_2+\cdots +{\sin }^{-1}{x}_{10}=5\pi$
We know that, ${\sin }^{-1}x\in [-\frac{\pi }{2},\frac{\pi }{2}]$
$\therefore$ The above equation is valid only when ${\sin }^{-1}{x}_i=\frac{\pi }{2}\Rightarrow x_i=1\therefore \sum _{i=1}^{10}{x}_i^2=10$