The angular elevation of the summit of a mountain is measured from three points on a straight level road. From a point due south of the summit the elevation is \(\alpha\), from a point due east of it the elevation is \(\beta\), and from the point of the road nearest to the summit the elevation is \(\gamma\).

If the direction of the road makes angle \(\theta\) east of north, prove that

\(\tan\theta=\tan\alpha\cot\beta\)

\(\tan^2\gamma=\tan^2\alpha+\tan^2\beta\).

Find \(\gamma\), if \(\theta=31^{\circ}\) and \(\alpha=8^{\circ}\).