Question

If $4{\sin }^{-1}x+{\cos }^{-1}x=\pi ,$ then find the value of $x$

Answer 1

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

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

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

Now we have,

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

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

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

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

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

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

$\Rightarrow x=\sin \left(\frac{\pi }{6}\right)=\frac{1}{2}$

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