Question

$2\tan h^{-1}\left(\frac{1}{\sqrt{2}}\right)=$

• A. $\cos h^{-1}\left(\frac{1}{3}\right)$
• B. $\cos h^{-1}\sqrt{3}$
• C. $\cos h^{-1}3$
• D. $\cos h^{-1}\frac{(\sqrt{2}+1)}{(\sqrt{2}–1)}$

Answer 1

The correct option is C

$\cos h^{-1}3$

Explanation for the correct option:

As we know that,

$\mathbf{2}\mathit{t}\mathit{a}\mathit{n}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{cos}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathbf{}\frac{\mathbf{(}\mathbf{1}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}}{\mathbf{(}\mathbf{1}\mathbf{}\mathbf{–}\mathbf{}{\mathbf{x}}^{\mathbf{2}}\mathbf{)}}\mathbf{;}\mathbf{}\mathbf{-}\mathbf{1}\mathbf{<}\mathit{x}\mathbf{<}\mathbf{1}$

where,

Put $x=\frac{1}{\sqrt{2}}$

$\Rightarrow 2\tan h^{-1}\left(\frac{1}{\sqrt{2}}\right)=\cos h^{-1}\left|\frac{\left(1+{\displaystyle \frac{1}{2}}\right)}{\left(1–{\displaystyle \frac{1}{2}}\right)}\right|$

$=\mathbf{cos}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathbf{3}$

Hence, Option ‘C’ is Correct.