Question

If $2{\sin }^{-1}x-{\cos }^{-1}x=\frac{\pi }{2}$, then $x=$

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

Answer 1

The correct option is D

$\frac{\surd 3}{2}$

Explanation for the correct option:

Step 1. Find the value of $x$:

Given that, $2{\sin }^{-1}x-{\cos }^{-1}x=\frac{\pi }{2}$ …..(1)

As we know,

$\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{2}}$ ……(2)

Step 2. Add equation (1) and (2):

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

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

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

$\Rightarrow$ $x=\sin \left(\frac{\mathrm{\pi }}{3}\right)$

$\therefore$ $x=\frac{\sqrt{3}}{2}$

Hence, Option ‘$D$’ is Correct.