Question

If ${\tan }^{-1}(x+1)+{\cot }^{-1}(x-1)={\sin }^{-1}\left(\frac{4}{5}\right)+{\cos }^{-1}\left(\frac{3}{5}\right)$, then $x$ has the value

• A. $4\sqrt{\frac{3}{7}}$
• B. $4\sqrt{\frac{7}{3}}$
• C. $14\sqrt{3}$
• D. $6\sqrt{7}$

Answer 1

Answer: (A)

${\tan }^{-1}\left(\frac{x+1+\frac{1}{x-1}}{1-\frac{x+1}{x-1}}\right)=2{\tan }^{-1}\left(\frac{4}{3}\right)=\pi +{\tan }^{-1}\left(\frac{-24}{7}\right)$
$\Rightarrow x^2=\frac{48}{7}\Rightarrow x=4\sqrt{\frac{3}{7}}$.