Answer
$cos \gamma=\pm \frac{1}{2}$
$\gamma=\frac{\pi}{3}$
or:
$\gamma=\frac{2\pi}{3}$
Work Step by Step
Given: $\alpha= \frac{\pi}{4}$,$\beta= \frac{\pi}{3}$
We need to find $\gamma$
$cos ^2\alpha+ cos ^2\beta+ cos^2 \gamma=1$
$cos ^2(\frac{\pi}{4})+cos ^2(\frac{\pi}{3})+cos^2 \gamma=1$
$\frac{1}{2}+\frac{1}{4}+cos^2 \gamma=1$
$cos^2 \gamma=\frac{1}{4}$
$cos \gamma=\pm \frac{1}{2}$
$\gamma=\frac{\pi}{3}$