Question

Solve $4{\sin }^{-1}x+{\cos }^{-1}x=\pi$

Answer 1

We know that ${\cos }^{-1}x+{\sin }^{-1}x=\frac{\pi }{2}$

$\Rightarrow {\cos }^{-1}x=\frac{\pi }{2}-{\sin }^{-1}x$

Now, $4{\sin }^{-1}x+\frac{\pi }{2}-{\sin }^{-1}x=\pi$

$\Rightarrow 4{\sin }^{-1}x-{\sin }^{-1}x=\pi -\frac{\pi }{2}=\frac{2\pi -\pi }{2}=\frac{\pi }{2}$

$\Rightarrow 3{\sin }^{-1}x=\frac{\pi }{2}$

$\Rightarrow {\sin }^{-1}x=\frac{\pi }{6}$

$\Rightarrow x=\sin \frac{\pi }{6}$

$\therefore x=\frac{1}{2}$