If ${\sin }^{-1}\frac{\alpha }{17}+{\cos }^{-1}\frac{4}{5}-{\tan }^{-1}\frac{77}{36}=0,0<\alpha <13,$  then ${\sin }^{-1}\left(\sin \alpha \right)+{\cos }^{-1}\left(\cos \alpha \right)$   is equal to $\frac{k\pi }{2}$ then  $k=$

Since,  ${\cos }^{-1}\frac{4}{5}={\tan }^{-1}\frac{3}{4}$

$$\begin{gathered} \Rightarrow {\sin }^{-1}\frac{\alpha }{17}={\tan }^{-1}\frac{77}{36}-{\tan }^{-1}\frac{3}{4}={\tan }^{-1}\left(\frac{\frac{77}{36}-\frac{3}{4}}{1+\frac{77}{36}.\frac{3}{4}}\right) \\ \Rightarrow {\sin }^{-1}\frac{\alpha }{17}={\tan }^{-1}\frac{8}{15}={\sin }^{-1}\frac{8}{17}\Rightarrow \frac{\alpha }{17}=\frac{8}{15}\Rightarrow \alpha =8 \\ Now, {\sin }^{-1}\left(\sin 8\right)+{\cos }^{-1}\left(\cos 8\right) =3\pi -8+8-2\pi =\pi \end{gathered}$$