Question

${\tanh }^{-1}\frac{1}{2}+{\coth }^{-1}3=$

• A. $\log \sqrt{6}$
• B. $\log 6$
• C. $-\log \sqrt{6}$
• D. $-\log 6$

Answer 1

Answer: (A)

$\tan h^{-1}\left(\frac{1}{2}\right)+\cot h^{-1}(3)$
$=\frac{1}{2}\ln \left(\frac{1+\frac{1}{2}}{1-\frac{1}{2}}\right)+\frac{1}{2}\ln \left(\frac{3+1}{3-1}\right)=\frac{1}{2}\log \left(\frac{\frac{3}{2}}{\frac{1}{2}}\right)+\frac{1}{2}\log \left(\frac{4}{2}\right)$
$=\frac{1}{2}\log 3+\frac{1}{2}\log 2$
$=\log \sqrt{3}+\log \sqrt{2}$
$=\log (\sqrt{3}\cdot \sqrt{2})=\log \sqrt{6}$