If x = tan 15^{circ},y = operatorname{cosec} | vedclass

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(A) First,calculate the value of $x$: $x = 	an 15^{\circ} = 	an(45^{\circ} - 30^{\circ}) = \frac{	an 45^{\circ} - 	an 30^{\circ}}{1 + 	an 45^{\ci

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$x < y < z$

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(A) First,calculate the value of $x$: $x = 	an 15^{\circ} = 	an(45^{\circ} - 30^{\circ}) = \frac{	an 45^{\circ} - 	an 30^{\circ}}{1 + 	an 45^{\circ} 	an 30^{\circ}} = \frac{1 - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} = \frac{(\sqrt{3} - 1)^2}{3 - 1} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \approx 2 - 1.732 = 0.268$. Next,calculate the value of $y$: $y = \operatorname{cosec} 75^{\circ} = \frac{1}{\sin(45^{\circ} + 30^{\circ})} = \frac{1}{\sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ}} = \frac{1}{\frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}}} = \frac{2\sqrt{2}}{\sqrt{3} + 1} = \sqrt{2}(\sqrt{3} - 1) = \sqrt{6} - \sqrt{2} \approx 2.449 - 1.414 = 1.035$. Finally,calculate the value of $z$: $z = 4 \sin 18^{\circ} = 4 \left( \frac{\sqrt{5} - 1}{4} ight) = \sqrt{5} - 1 \approx 2.236 - 1 = 1.236$. Comparing the values: $0.268 < 1.035 < 1.236$,which implies $x < y < z$.

If $x = \tan 15^{\circ}$,$y = \operatorname{cosec} 75^{\circ}$ and $z = 4 \sin 18^{\circ}$,then :

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