Prove that cos-112/13 + sin-1 3/5 …

Let {tex}{\cos ^{ - 1}}\frac{{12}}{{13}} = \theta{/tex} so that {tex}\cos \theta = \frac{{12}}{{13}}{/tex}
{tex}\therefore \sin \theta = \sqrt {1 - {{\cos }^2}\theta } = \sqrt {1 - \frac{{144}}{{169}}}{/tex}{tex}= \sqrt {\frac{{25}}{{169}}} = \frac{5}{{13}}{/tex}
Again, Let {tex}{\sin ^{ - 1}}\frac{3}{5} = \phi{/tex} so that {tex}\sin \phi = \frac{3}{5}{/tex}
{tex}\therefore \cos \phi = \sqrt {1 - {{\sin }^2}\phi } = \sqrt {1 - \frac{9}{{25}}} = \sqrt {\frac{{16}}{{25}}} = \frac{4}{5}{/tex}
Since {tex}\sin \left( {\theta + \phi } \right) = \sin \theta \cos \phi + \cos \theta \sin \phi {/tex} {tex} = \frac{5}{{13}} \times \frac{4}{5} + \frac{{12}}{{13}} \times \frac{3}{5}{/tex} 
{tex}= \frac{{20 + 36}}{{65}} = \frac{{56}}{{65}}{/tex}
{tex}\Rightarrow \theta + \phi = {\sin ^{ - 1}}\frac{{56}}{{65}}{/tex}
{tex} \Rightarrow {\cos ^{ - 1}}\frac{{12}}{{13}} + {\sin ^{ - 1}}\frac{3}{5} = {\sin ^{ - 1}}\frac{{56}}{{65}}{/tex}