Question

The value of ${\sin }^{-1}\left(\cos 2\right)-{\cos }^{-1}\left(\sin 2\right)+{\tan }^{-1}\left(\cot 4\right)-{\cot }^{-1}\left(\tan 4\right)+{\sec }^{-1}\left(\csc 6\right)-{\csc }^{-1}\left(\sec 6\right)=$

• A. $0$
• B. $3\pi$
• C. $8-3\pi$
• D. $5\pi -16$

Answer 1

The correct option is D $5\pi -16$

${\sin }^{-1}\left(\cos 2\right)-{\cos }^{-1}\left(\sin 2\right)+{\tan }^{-1}\left(\cot 4\right)-{\cot }^{-1}\left(\tan 4\right)+{\sec }^{-1}\left(\csc 6\right)-{\csc }^{-1}\left(\sec 6\right)$

$=\{\frac{\pi }{2}-{\cos }^{-1}\left(\cos 2\right)\}-\{\frac{\pi }{2}-{\sin }^{-1}\left(\sin 2\right)\}+\{\frac{\pi }{2}-{\cot }^{-1}\left(\cot 4\right)\}-\{\frac{\pi }{2}-{\tan }^{-1}\left(\tan 4\right)\}+\{\frac{\pi }{2}-{\csc }^{-1}\left(\csc 6\right)\}$
$-\{\frac{\pi }{2}-{\sec }^{-1}\left(\sec 6\right)\}$

$={\sin }^{-1}\left(\sin 2\right)-{\cos }^{-1}\left(\cos 2\right)+{\tan }^{-1}\left(\tan 4\right)-{\cot }^{-1}\left(\cot 4\right)+{\sec }^{-1}\left(\sec 6\right)-{\csc }^{-1}\left(\csc 6\right)$

$=(\pi -2)-2+(4-\pi )-(4-\pi )+(2\pi -6)-(6-2\pi )$

$=5\pi -16$