We know that, if a line makes an angle $\alpha , \beta \& \gamma$ with positive direction of $X-\text{axis}, Y-\text{axis} \& Z-\text{axis}$ then

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1 ..\left(i\right)$

Now, given angles are $\alpha ={\tan }^{-1}\sqrt{7}$ and $\beta ={\tan }^{-1}\left(\sqrt{\frac{5}{3}}\right)$.

Then, 

$\cos \left({\tan }^{-1}\left(\sqrt{7}\right)\right)=\cos \left({\cos }^{-1}\left(\frac{1}{2\sqrt{2}}\right)\right)=\frac{1}{2\sqrt{2}}$

$\cos \left({\tan }^{-1}\left(\frac{\sqrt{5}}{\sqrt{3}}\right)\right)=\cos \left({\cos }^{-1}\left(\frac{\sqrt{3}}{2\sqrt{2}}\right)\right)=\frac{\sqrt{3}}{2\sqrt{2}}$

Therefore, from $\left(i\right)$, we get

$\Rightarrow {\left(\frac{1}{2\sqrt{2}}\right)}^2+{\left(\frac{\sqrt{3}}{2\sqrt{2}}\right)}^2+{\cos }^2\gamma =1$

$\Rightarrow \frac{1}{8}+\frac{3}{8}+{\cos }^2\gamma =1$

$\Rightarrow {\cos }^2\gamma =1-\frac{1}{2}$

$\Rightarrow {\cos }^2\gamma =\frac{1}{2}$

$\Rightarrow \cos \gamma =\pm \frac{1}{\sqrt{2}}$

$\Rightarrow \gamma =\frac{\mathrm{\pi }}{4} \text{or} \frac{3\mathrm{\pi }}{4}$.