If a line makes angles ${\cos }^{-1}\left(\frac{1}{5}\right), {\cos }^{-1}\left(\frac{3}{5}\right)$ with X and Y-axes respectively, then the angle made by the line with Z-axis is

We know that if $\alpha ,\beta ,\gamma$ are the angles made by the line with axes then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

Hence, with the given data 

$\begin{array}{rcl}\frac{1}{25}+\frac{9}{25}+x& =& 1\\ x& =& 1-\frac{10}{25}\\ & =& \frac{15}{25}\\ & =& \frac{3}{5}\end{array}$

Therefore, the angle made by the line with $z-$axis is ${\cos }^{-1}\left(\sqrt{\frac{3}{5}}\right)$