$\begin{align}  & \text{Let }{{\sin }^{-1}}\left( \frac{24}{25} \right)=\theta \text{ and }{{\cos }^{-1}}\left( \frac{3}{5} \right)=\alpha  \\ & \sin \theta =\frac{24}{25},\text{ } \\ & \cos \theta =\sqrt{1-{{\sin }^{2}}\theta }=\sqrt{1-{{\left( \frac{24}{25} \right)}^{2}}}=\frac{7}{25} \\ & \tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\tfrac{24}{25}}{\tfrac{7}{25}}=\frac{24}{7} \\ & \theta ={{\tan }^{-1}}\frac{24}{7}={{\sin }^{-1}}\frac{24}{25}={{\cos }^{-1}}\frac{7}{25} \\ & \text{Also }\cos \alpha =\frac{3}{5}, \\ & \sin \alpha =\sqrt{1-{{\cos }^{2}}\alpha }=\sqrt{1-\left( {{\frac{3}{5}}^{2}} \right)}=\frac{4}{5} \\ & \tan \alpha =\frac{\sin \alpha }{\cos \alpha }=\frac{\tfrac{4}{5}}{\tfrac{3}{5}}=\frac{4}{3} \\ & \alpha ={{\sin }^{-1}}\left( \frac{4}{5} \right)={{\cos }^{-1}}\left( \frac{3}{5} \right)={{\tan }^{-1}}\left( \frac{24}{7} \right) \\ & \text{L}\text{.H}\text{.S} \\ & {{\sin }^{-1}}(\tfrac{24}{25})+2{{\cos }^{-1}}(\tfrac{3}{5})={{\cos }^{-1}}\left[ \cos \left( {{\sin }^{-1}}\tfrac{24}{25}+2{{\cos }^{-1}}\tfrac{3}{5} \right) \right] \\ & ={{\cos }^{-1}}\left[ \cos ({{\sin }^{-1}}\tfrac{24}{25})\cos (2{{\cos }^{-1}}\tfrac{3}{5})-\sin ({{\sin }^{-1}}\tfrac{24}{25})\sin (2{{\cos }^{-1}}\tfrac{3}{5}) \right] \\ & ={{\cos }^{-1}}\left[ \cos ({{\cos }^{-1}}\tfrac{7}{25})\cos (2{{\cos }^{-1}}\tfrac{3}{5})-\sin ({{\sin }^{-1}}\tfrac{24}{25})\sin (2{{\cos }^{-1}}\tfrac{3}{5}) \right] \\ & ={{\cos }^{-1}}\left[ \tfrac{7}{25}\cos [2{{\cos }^{-1}}(\tfrac{3}{5})]-\tfrac{24}{25}\sin [(2{{\sin }^{-1}}\tfrac{4}{5})] \right] \\ & ={{\cos }^{-1}}\left[ \tfrac{7}{25}\times 2{{\cos }^{2}}{{\cos }^{-1}}\tfrac{3}{5})-1-\tfrac{24}{25}\times 2\sin ({{\sin }^{-1}}\tfrac{4}{5})\cos ({{\sin }^{-1}}\tfrac{4}{5}) \right] \\ & ={{\cos }^{-1}}\left[ \tfrac{7}{25}\times 2{{\cos }^{2}}{{\cos }^{-1}}\tfrac{3}{5})-1-\tfrac{24}{25}\times 2\sin ({{\sin }^{-1}}\tfrac{4}{5})\cos ({{\cos }^{-1}}\tfrac{3}{5}) \right] \\ & ={{\cos }^{-1}}\left[ \tfrac{7}{25}\times 2\times \tfrac{9}{25}-1-\tfrac{24}{25}(2\times \tfrac{4}{5}\times \tfrac{3}{5}) \right]={{\cos }^{-1}}\left[ -\tfrac{49}{625}-\tfrac{576}{625} \right] \\ & {{\sin }^{-1}}(\tfrac{24}{25})+2{{\cos }^{-1}}(\tfrac{3}{5})={{\cos }^{-1}}(-1)=\pi \text{ } \\\end{align}$