The values of $sin \left(cos^{-1}\frac{3}{5}+cosec^{-1}\frac{13}{5}\right)$, is

$\frac{63}{65}$

$sin \left(cos^{-1}\frac{3}{5}+cosec^{-1}\frac{13}{5}\right)$

$=sin \left(cos^{-1}\frac{3}{5}\right)cos\left(cosec^{-1}\frac{13}{5}\right)+cos\left(cos^{-1}\frac{3}{5}\right)sin\left(cosec^{-1}\frac{13}{5}\right)$

$= sin\left(sin^{-1}\frac{4}{5}\right)cos\left(cos^{-1}\frac{12}{13}\right)+cos\left(cos^{-1}\frac{3}{5}\right)sin\left(sin^{-1}\frac{5}{13}\right)$

$=\frac{4}{5}×\frac{12}{13}+\frac{3}{5}×\frac{5}{13}=\frac{63}{65}$