Question

$\cos \left[{\sin }^{-1}\left(\frac{5}{13}\right)\right]=$

• A. $\frac{12}{13}$
• B. $-\frac{12}{13}$
• C. $\frac{5}{12}$
• D. None of these

Answer 1

The correct option is A

$\frac{12}{13}$

Explanation for the correct option:

Given, $\cos \left[{\sin }^{-1}\left(\frac{5}{13}\right)\right]$

$=\cos \left[{\cos }^{-1}\sqrt{1-{\left(\frac{5}{13}\right)}^2}\right]$ ; $\left[\mathbf{\because }\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\right]$

$=\cos \left[{\cos }^{-1}\left(\sqrt{1-\frac{25}{169}}\right)\right]$

$=\cos \left[{\cos }^{-1}\left(\sqrt{\frac{144}{169}}\right)\right]$

$=\cos \left[{\cos }^{-1}\left(\frac{12}{13}\right)\right]$

$=\frac{\mathbf{12}}{\mathbf{13}}$

Hence, Option ‘A’ is Correct.