Given,

${\sin }^{-1}\left(\frac{\alpha }{17}\right)+{\cos }^{-1}\left(\frac{4}{5}\right)-{\tan }^{-1}\left(\frac{77}{36}\right)=0$

$\Rightarrow {\sin }^{-1}\left(\frac{\alpha }{17}\right)={\tan }^{-1}\left(\frac{77}{36}\right)-{\cos }^{-1}\left(\frac{4}{5}\right)$

$\Rightarrow {\sin }^{-1}\left(\frac{\alpha }{17}\right)={\tan }^{-1}\left(\frac{77}{36}\right)-{\tan }^{-1}\left(\frac{3}{4}\right)$

$\Rightarrow {\sin }^{-1}\left(\frac{\alpha }{17}\right)={\tan }^{-1}\left(\frac{\frac{77}{36}-\frac{3}{4}}{1+\frac{77}{36}\times \frac{3}{4}}\right)$

$\Rightarrow {\sin }^{-1}\left(\frac{\alpha }{17}\right)={\tan }^{-1}\left(\frac{8}{15}\right)$

$\Rightarrow {\sin }^{-1}\left(\frac{\alpha }{17}\right)={\sin }^{-1}\left(\frac{8}{17}\right)$

$\Rightarrow \alpha =8$

Now solving $$\begin{gathered} {\sin }^{-1}\left(\sin \alpha \right)+{\cos }^{-1}\left(\cos \alpha \right) \\ ={\sin }^{-1}\left(\sin \alpha \right)+{\cos }^{-1}\left(\cos \alpha \right) \end{gathered}$$

$\Rightarrow {\sin }^{-1}\left(\sin 8\right)+{\cos }^{-1}\left(\cos 8\right)=3\mathrm{\pi }-8+8-2\mathrm{\pi }=\mathrm{\pi }$