Question

The value of $\cos \left[{\tan }^{-1}\right(\tan 4\left)\right]$ is

• A. $\frac{1}{\sqrt{17}}$
• B. $-\frac{1}{\sqrt{17}}$
• C. $\cos 4$
• D. $-\cos 4$

Answer 1

The correct option is D $-\cos 4$

We know that,
${\tan }^{-1}\left(\tan x\right)=x-\pi ;\frac{\pi }{2}<x<\frac{3\pi }{2}\Rightarrow {\tan }^{-1}\left(\tan 4\right)=4-\pi \Rightarrow \cos \left[{\tan }^{-1}\right(\tan 4\left)\right]=\cos (4-\pi )\Rightarrow \cos \left[{\tan }^{-1}\right(\tan 4\left)\right]=\cos (\pi -4)\therefore \cos \left[{\tan }^{-1}\right(\tan 4\left)\right]=-\cos 4$