Question

$\sin h^{-1}2+{\cosh }^{-1}2-{\tanh }^{-1}\frac{2}{3}+{\text{coth}}^{-1}(-2)=$

• A. $\log \left(\frac{4+2\sqrt{3}+2\sqrt{5}+\sqrt{15}}{\sqrt{15}}\right)$
• B. $\log \left(\frac{4+\sqrt{3}+\sqrt{5}+\sqrt{15}}{\sqrt{15}}\right)$
• C. $\log \frac{(2+\sqrt{3})(2+\sqrt{5})\sqrt{5}}{\sqrt{3}}$
• D. $\log \frac{(2+\sqrt{3})(2+\sqrt{5})\sqrt{3}}{\sqrt{5}}$

Answer 1

Answer: (D)

We have,
${\sinh }^{-1}(2)+{\cosh }^{-1}(2)-{\tanh }^{-1}\left(\frac{2}{3}\right)+\cot h^{-1}(-2)$
$=\ln \left(2+\sqrt{2^2+1}\right)+\ln \left(2+\sqrt{2^2-1}\right)$
$-\frac{1}{2}\ln \left(\frac{1+2/3}{1-2/3}\right)+\frac{1}{2}\ln \left(\frac{-2+1}{-2-1}\right)$
$=\ln (2+\sqrt{5})+\ln (2+\sqrt{3})-\frac{1}{2}\ln 5+\frac{1}{2}\ln \frac{1}{3}$
$=\ln \left[\frac{(2+\sqrt{5})(2+\sqrt{3})(\sqrt{3})}{\sqrt{5}}\right]$
$[\because {\sinh }^{-1}x=\ln \left(x+\sqrt{x^2+1}\right),$,
${\cosh }^{-1}x=\ln \left(x+\sqrt{x^2-1}\right),{\tanh }^{-1}x=\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)$,
${\text{coth}}^{-1}x=\frac{1}{2}\ln \left(\frac{x+1}{x-1}\right)]$