$\Delta XYZ$ is right-angled at $Y$. If $\cos X = \frac{3}{5}$,then what is the value of $\operatorname{cosec} Z$?

If $x = \operatorname{cosec} \theta - \sin \theta$ and $y = \sec \theta - \cos \theta$,then the relation between $x$ and $y$ is

If $\cos \theta + \sin \theta = \sqrt{2} \cos \theta$,then $\cos \theta - \sin \theta =$

If $\cot \theta + \cos \theta = p$ and $\cot \theta - \cos \theta = q$,then $(p^2 - q^2)^2$ in terms of $p$ and $q$ is: (in $pq$)

$A$ 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 $ft$)

If $\tan \theta = \sqrt{\frac{3}{2}}$,the sum of the infinite series $1 + 2(1 - \cos \theta) + 3(1 - \cos \theta)^2 + 4(1 - \cos \theta)^3 + \dots \infty$ is