Question

The value of ${\tanh }^{-1}\left(2^{-1}\right)$ is

• A. $\log \left(2\right)$
• B. $\log \left(2^{-1}\right)$
• C. $\log \left(\sqrt{3}\right)$
• D. None of these.

Answer 1

The correct option is C

$\log \left(\sqrt{3}\right)$

Explanation for the correct option

The given trigonometric expression: ${\tanh }^{-1}\left(2^{-1}\right)$.

It is known that ${\mathbf{tanh}}^{\mathbf{-}\mathbf{1}}\left(x\right)\mathbf{=}\frac{\mathbf{log}\left[{\displaystyle \frac{1+x}{1-x}}\right]}{\mathbf{2}}$.

Thus, ${\tanh }^{-1}\left(2^{-1}\right)=\frac{\log \left[{\displaystyle \frac{1+2^{-1}}{1-2^{-1}}}\right]}{2}$.

$$\begin{gathered} \Rightarrow {\tanh }^{-1}\left(2^{-1}\right)=\frac{\log \left[{\displaystyle \frac{1+{\displaystyle \frac{1}{2}}}{1-{\displaystyle \frac{1}{2}}}}\right]}{2} \\ \Rightarrow {\tanh }^{-1}\left(2^{-1}\right)=\frac{\log \left[{\displaystyle \frac{{\displaystyle \frac{3}{2}}}{{\displaystyle \frac{1}{2}}}}\right]}{2} \\ \Rightarrow {\tanh }^{-1}\left(2^{-1}\right)=\frac{\log \left(3\right)}{2} \\ \Rightarrow {\tanh }^{-1}\left(2^{-1}\right)=\log {\left(3\right)}^{\frac{1}{2}} \\ \Rightarrow {\tanh }^{-1}\left(2^{-1}\right)=\log \left(\sqrt{3}\right) \end{gathered}$$

Hence, option C is correct .