Question

The value of ${\sin }^{-1}(\cos 2)-{\cos }^{-1}(\sin 2)+{\tan }^{-1}(\cot 4)-{\cot }^{-1}(\tan 4)+{\sec }^{-1}(\mathrm{cosec}6)-{\mathrm{cosec}}^{-1}(\sec 6)$ is

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

Answer 1

Answer: (D)

Given expression $=\left\{\frac{\pi }{2}-{\cos }^{-1}(\cos 2)\right\}-\left\{\frac{\pi }{2}-{\sin }^{-1}(\sin 2)\right\}$
$+\left\{\frac{\pi }{2}-{\cot }^{-1}(\cot 4)\right\}-\left\{\frac{\pi }{2}-{\tan }^{-1}(\tan 4)\right\}+\left\{\frac{\pi }{2}-{\mathrm{cosec}}^{-1}(\mathrm{cosec}6)\right\}-\left\{\frac{\pi }{2}-{\sec }^{-1}(\sec 6)\right\}$
$={\sin }^{-1}(\sin 2)-{\cos }^{-1}(\cos 2)+{\tan }^{-1}(\tan 4)-{\cot }^{-1}(\cot 4)+{\sec }^{-1}(\sec 6)-{\mathrm{cosec}}^{-1}(\mathrm{cosec}6)$
$=(\pi -2)-2+(4-\pi )-(4-\pi )+(2\pi -6)-(6-2\pi )=\pi -4+4\pi -12=5\pi -16$