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

$E=\left\{\frac{\pi }{2}-{\cos }^{-1}\left(\cos 2\right)\right\}-\left\{\frac{\pi }{2}-{\sin }^{-1}\left(\sin 2\right)\right\}+\left\{\frac{\pi }{2}-{\cot }^{-1}\left(\cot 4\right)\right\}$ 
$-\left\{\frac{\pi }{2}-{\tan }^{-1}(\tan 4)\right\}+\left\{\frac{\pi }{2}-cose{c}^{-1}(cosec6)\right\}+\left\{\frac{\pi }{2}-{\sec }^{-1}(\sec 6)\right\}$ 
$=-2+(\pi -2)-(4-\pi )+(4-\pi )-(6-2\pi )+(2\pi -6)$ 
$=5\pi -16$