cosec$$\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right]$$ is equal to :

$$\cos ec\left( {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right)$$<br><br>$$\cos ec\left( {2{{\tan }^{ - 1}}\left( {{1 \over 5}} \right) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right)$$<br><br>$$ = \cos ec\left( {{{\tan }^{ - 1}}\left( {{{2\left( {{1 \over 5}} \right)} \over {1 - {{\left( {{1 \over 5}} \right)}^2}}}} \right) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right)$$<br><br>$$ = \cos ec\left( {{{\tan }^{ - 1}}\left( {{5 \over {12}}} \right) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right)$$<br><br>Let $${\tan ^{ - 1}}(5/12) = \theta \Rightarrow \sin \theta = {5 \over {13}},\cos \theta = {{12} \over {13}}$$<br><br>and $${\cos ^{ - 1}}\left( {{4 \over 5}} \right) = \phi \Rightarrow \cos \phi = {4 \over 5}$$ and $\sin \phi = {3 \over 5}$<br><br>$= \cos ec(\theta + \phi )$<br><br>$= {1 \over {\sin \theta \cos \phi + \cos \theta \sin \phi }}$<br><br>$$ = {1 \over {{5 \over {13}}.{4 \over 5} + {{12} \over {13}}.{3 \over 5}}} = {{65} \over {56}}$$