Explanation

(a)
$x = \tan{15{^\circ}}$
$= \frac{\tan{45{^\circ} - \tan{30{^\circ}}}}{\tan{45{^\circ}.\tan{30{^\circ}}}}$
$= \frac{\sqrt{3} - 1}{\sqrt{3} + 1} = \frac{\left( \sqrt{3} - 1 \right)^{2}}{3 - 1} = 2 - \sqrt{3}$
And $y = cosec\ 75{^\circ} = \frac{1}{\sin{(45{^\circ} + 30{^\circ})}}$
$= \frac{1}{\sin{45{^\circ}\cos{30{^\circ} + \cos{45{^\circ}\sin{30{^\circ}}}}}}$
$= \frac{1}{\frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}}} = \frac{2\sqrt{2}}{\sqrt{3} + 1} \times \frac{\sqrt{3} - 1}{\sqrt{3} - 1} = \sqrt{6} - \sqrt{2}$
And $z = 4\sin{18{^\circ} = 4\left( \frac{\sqrt{5} - 1}{4} \right)} = \sqrt{5} - 1$
It is clear from above that
$\left( 2 - \sqrt{3} \right) < \left( \sqrt{6} - \sqrt{2} \right) < (\sqrt{5} - 1)$
$\Rightarrow \ \ \ \ x < y < z$