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

$\begin{array}{l}{\sin }^{-1}(\cos 2)=\frac{\mathrm{\pi }}{2}-{\cos }^{-1}(\cos 2)=\frac{\mathrm{\pi }}{2}-2{\cos }^{-1}(\sin 2)=\frac{\mathrm{\pi }}{2}-{\sin }^{-1}(\sin 2)=\frac{\mathrm{\pi }}{2}-{\sin }^{-1}(\sin (\mathrm{\pi }-2\left)\right)=\frac{\mathrm{\pi }}{2}-\mathrm{\pi }+2=2-\frac{\mathrm{\pi }}{2}\\ {\tan }^{-1}(\cot 4)=\frac{\mathrm{\pi }}{2}-{\cot }^{-1}(\cot (-4\left)\right)=\frac{\mathrm{\pi }}{2}-{\cot }^{-1}(\cot (4-\mathrm{\pi }\left)\right)=\frac{\mathrm{\pi }}{2}-(4-\mathrm{\pi })=\frac{3\mathrm{\pi }}{2}-4\\ {\cot }^{-1}(\tan 4)=\frac{\mathrm{\pi }}{2}-{\tan }^{-1}(\tan (4\left)\right)=\frac{\mathrm{\pi }}{2}-{\tan }^{-1}(\tan (4-\mathrm{\pi }\left)\right)\frac{\mathrm{\pi }}{2}-(4-\mathrm{\pi })=\frac{3\mathrm{\pi }}{2}-4\\ {\sec }^{-1}(\mathrm{cosec}6)=\frac{\mathrm{\pi }}{2}-\mathrm{cosec}(\mathrm{cosec}6)=\frac{\mathrm{\pi }}{2}-{\mathrm{cosec}}^{-1}(\mathrm{cosec}(6-2\mathrm{\pi }\left)\right)\\ =\frac{\mathrm{\pi }}{2}-(6-2\mathrm{\pi })=\frac{5\mathrm{\pi }}{2}-6\mathrm{cosec}(\sec 6)=\frac{\mathrm{\pi }}{2}-{\sec }^{-1}(\sec 6)=\frac{\mathrm{\pi }}{2}-{\sec }^{-1}(\sec (2\mathrm{\pi }-6\left)\right)\\ =\frac{\mathrm{\pi }}{2}-(2\mathrm{\pi }-6)=\frac{-3\mathrm{\pi }}{2}+6\\ {\sin }^{-1}(\cos 2)-{\cos }^{-1}(\sin 2)+{\tan }^{-1}(\cot 4)-{\cot }^{-1}(\tan 4)+{\sec }^{-1}(\mathrm{cosec}6)-{\mathrm{cosec}}^{-1}(\sec 6)\\ =\frac{\mathrm{\pi }}{2}-2-2+\frac{\mathrm{\pi }}{2}+\frac{3\mathrm{\pi }}{2}-4-\frac{3\mathrm{\pi }}{2}+4+\frac{5\mathrm{\pi }}{2}-6+\frac{3\mathrm{\pi }}{2}-6=5\mathrm{\pi }-16\end{array}$