Question

Which one of the following quantities is negative?

• A. $\cos \left({\tan }^{-1}\right(\tan 4\left)\right)$
• B. $\sin \left({\cot }^{-1}\right(\cot 4\left)\right)$
• C. $\tan \left({\cos }^{-1}\right(\cos 5\left)\right)$
• D. $\cot \left({\sin }^{-1}\right(\sin 4\left)\right)$

Answer 1

The correct option is D $\cot \left({\sin }^{-1}\right(\sin 4\left)\right)$

We will solve this by options
Option A, $\cos \left({\tan }^{-1}\right(\tan 4\left)\right)$
$\cos \left({\tan }^{-1}\right(\tan 4\left)\right)=\cos \left({\tan }^{-1}\right(\tan (\pi +(4-\pi \left)\right))=\cos ({\tan }^{-1}\left(\tan \right(4-\pi \left)\right))=\cos (4-\pi )$ (which is positive)
Option B,$\sin \left({\cot }^{-1}\right(\cot 4\left)\right)$
$\cos \left({\tan }^{-1}\right(\tan 4\left)\right)=\sin \left({\cot }^{-1}\right(\cot (\pi +(4-\pi \left)\right))=\sin ({\cot }^{-1}\left(\cot \right(4-\pi \left)\right))=\sin (4-\pi )$ (which is positive)
$\tan \left({\cos }^{-1}\right(\cos 5\left)\right)=\tan \left({\cos }^{-1}\right(\cos (2\pi -(2\pi -5\left)\right)))=\tan \left({\cos }^{-1}\right(\cos (2\pi -5)\left)\right)=\tan (2\pi -5)$ (which is positive)
$\cot \left({\sin }^{-1}\right(\sin 4\left)\right)=\cot \left({\sin }^{-1}\right(\sin (\pi +(4-\pi \left)\right))=\cot ({\sin }^{-1}(-\sin (4-\pi \left)\right))=\cot (\pi -4)$ (which is negative)