Answer
The proof is below.
Work Step by Step
We know that the force of friction is equal to the coefficient of friction times the normal force. We know that the normal force is given by $mgcos\theta$. Thus, we find:
$F_f =mgcos\theta\mu $
We have to find the distance the object travels, which is:
$x=\frac{\sqrt{2hR-h^2}}{cos\theta}$
We know that work is equal to force times distance, so we find:
$W = mgcos\theta\mu \times \frac{\sqrt{2hR-h^2}}{cos\theta}$
$W=mg\mu \sqrt{2hR-h^2}$
