Question

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

• A. $\frac{1}{\sqrt{5}}$
• B. $-\frac{1}{\sqrt{5}}$
• C. $\cos \left(2\right)$
• D. $-\cos \left(2\right)$

Answer 1

The correct option is D

$-\cos \left(2\right)$

Explanation for correct option

Given trigonometric function is $\cos \left({\tan }^{-1}\left(\tan \left(2\right)\right)\right)$

$$\begin{gathered} \cos \left({\tan }^{-1}\left(\tan \left(2\right)\right)\right) \\ =\cos \left({\tan }^{-1}\left(-\tan \left(\pi -2\right)\right)\right) \\ =\cos \left(-\pi +2\right) \\ =\cos \left(-\left(\pi -2\right)\right) \\ =\cos \left(\pi -2\right) \\ =-\cos \left(2\right) \end{gathered}$$

Hence the correct option is $\mathrm{Option}\left(\mathrm{D}\right)$