Question

If $\sin {}^{-1}\frac{1}{x}=2{\tan }^{-1}\frac{1}{7}+{\cos }^{-1}\frac{3}{5}$, then $x=$ ___

• A. $\frac{24}{117}$
• B. $\frac{7}{3}$
• C. $\frac{125}{117}$
• D. None of these

Answer 1

The correct option is C $\frac{125}{117}$

$2{\tan }^{-1}\left(\frac{1}{7}\right)+{\cos }^{-1}\left(\frac{3}{5}\right)$

$={\tan }^{-1}\left(\frac{1}{7}\right)+{\tan }^{-1}\left(\frac{1}{7}\right)+{\cos }^{-1}\left(\frac{3}{5}\right)$

$={\tan }^{-1}\left(\frac{\frac{1}{7}+\frac{1}{7}}{1-\frac{1}{7}\times \frac{1}{7}}\right)+{\cos }^{-1}\left(\frac{3}{5}\right)$ $[\because {\tan }^{-1}+{\tan }^{-1}y{\tan }^{-1x}={\tan }^{-1}\left(\frac{x+y}{1-xy}\right)]$

$={\tan }^{-1}\left(\frac{\frac{2}{7}}{1-\frac{1}{49}}\right)+\theta$

$={\tan }^{-1}\left(\frac{7}{24}\right)+{\tan }^{-1}\left(\frac{4}{3}\right)$

$={\tan }^{-1}\left(\frac{\frac{7}{24}+\frac{4}{3}}{1-\frac{7}{24}\times \frac{4}{3}}\right)$

$={\tan }^{-1}\left(\frac{117}{44}\right)$

$\because {\sin }^{-1}\left(\frac{1}{x}\right)={\tan }^{-1}\left(\frac{117}{44}\right)$

Let $\theta ={\tan }^{-1}\left(\frac{117}{44}\right)$

$\therefore \tan \theta =\frac{117}{44}$

$\therefore \sin \theta =\frac{117}{\sqrt{{117}^2+{44}^2}}=\frac{117}{125}$

$\therefore \theta ={\sin }^{-1}\left(\frac{117}{125}\right)$

$\therefore {\sin }^{-1}\left(\frac{1}{x}\right)={\sin }^{-1}\left(\frac{117}{125}\right)$

On comparing

$\frac{1}{x}=\frac{117}{125}$

$\therefore x=\frac{125}{117}$