If $x = r \cos \theta \cos \phi$,$y = r \cos \theta \sin \phi$,and $z = r \sin \theta$,then the value of $x^{2} + y^{2} + z^{2}$ is

$\cos 2\theta + 2\cos \theta$ is always

If $\tan \theta + \cot \theta = 5,$ then $\tan^2 \theta + \cot^2 \theta$ is

If $\cos(\alpha + \beta) = \frac{3}{5}$,$\sin(\alpha - \beta) = \frac{5}{13}$ and $0 < \alpha, \beta < \frac{\pi}{4}$,then $\tan(2\alpha)$ is equal to

If $f(x) = \cos^2 x + \sec^2 x$,then its value is always:

$A$ ladder is placed along a wall such that its upper end is touching the top of the wall. The foot of the ladder is $10 \, ft$ away from the wall and the ladder makes an angle of $60^{\circ}$ with the ground. When a man starts climbing on it,it slips and now the ladder makes an angle of $30^{\circ}$ with the ground. How much did the ladder slip down from the top of the wall? (in $ft$)