Question

If $3{\cos }^{-1}x+{\sin }^{-1}x=\pi$, then $x=.....$

• A. $\frac{\sqrt{3}}{2}$
• B. $-\frac{1}{\sqrt{2}}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{2}$

Answer 1

The correct option is C $\frac{1}{\sqrt{2}}$

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

$3{\cos }^{-1}x+{\sin }^{-1}x=2{\cos }^{-1}x+{\cos }^{-1}x+{\sin }^{-1}x=\pi$

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

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

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

$x=\cos \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$