Question

$\cos \left[2{\cos }^{-1}\left(\frac{1}{5}\right)+{\sin }^{-1}\left(\frac{1}{5}\right)\right]=?$

• A. $\frac{2\sqrt{6}}{5}$
• B. $\frac{-2\sqrt{6}}{5}$
• C. $\frac{1}{5}$
• D. $\frac{-1}{5}$

Answer 1

The correct option is B

$\frac{-2\sqrt{6}}{5}$

Explanation for the correct option:

Find the value of given expression:

$\cos \left[2{\cos }^{-1}\left(\frac{1}{5}\right)+{\sin }^{-1}\left(\frac{1}{5}\right)\right]$
$=\cos \left[{\cos }^{-1}\left(\frac{1}{5}\right)+{\cos }^{-1}\left(\frac{1}{5}\right)+{\sin }^{-1}\left(\frac{1}{5}\right)\right]$
$=\cos \left[{\cos }^{-1}(\frac{1}{5})+\frac{\pi }{2}\right]$ ; $\left[\mathbf{\because }\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathit{\theta }\mathbf{+}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\mathit{\theta }\mathbf{=}\frac{\mathbf{\pi }}{\mathbf{2}}\right]$
$=-\sin \left[{\cos }^{-1}\left(\frac{1}{5}\right)\right]$ ; $\left[\mathbf{\because }\mathit{c}\mathit{o}\mathit{s}\left(\frac{\pi }{2}+\theta \right)\mathbf{=}\mathbf{-}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\right]$

$=-\sin \left[{\sin }^{-1}\left(\sqrt{1-{\left(\frac{1}{5}\right)}^2}\right)\right]$ ; $\left[\mathbf{\because }\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\mathit{\theta }\mathbf{=}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{\theta }}^{\mathbf{2}}}\right]$

$=-\sqrt{1-{\left(\frac{1}{5}\right)}^2}$

$=-\frac{\sqrt{24}}{5}$

$=\frac{\mathbf{-}\mathbf{2}\sqrt{\mathbf{6}}}{\mathbf{5}}$​

Hence, Option ‘B’ is Correct.​​