$\cos \left(2{\tan }^{-1}\frac{1}{5}+{\cos }^{-1}\frac{63}{65}\right)=$

$=\cos \left[{\cos }^{-1}\frac{24}{26}+{\cos }^{-1}\frac{63}{65}\right]=\cos \left[{\cos }^{-1}\frac{12}{13}+{\cos }^{-1}\frac{63}{65}\right]$

$=\cos \left[\pi -{\cos }^{-1}\left(\sqrt{1-{\left(\frac{12}{13}\right)}^2}.\sqrt{1-{\left(\frac{63}{65}\right)}^2}-\frac{12}{13}.\frac{63}{65}\right)\right]$

$=-\cos \left[{\cos }^{-1}\frac{5}{13}.\frac{16}{65}-\frac{12}{13}.\frac{63}{65}\right]=-\cos \left[{\cos }^{-1}\left(\frac{80-756}{845}\right)\right]$

$=\cos \left[{\cos }^{-1}\left(-\frac{676}{845}\right)\right]=-\cos \left[\pi -{\cos }^{-1}\left(\frac{676}{845}\right)\right]$

$=\cos \left({\cos }^{-1}\frac{52}{65}\right)=\cos \left({\cos }^{-1}\frac{4}{5}\right)=\frac{4}{5}$