Question

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

• A. $x<y<z$
• B. $y<z<x$
• C. $z<x<y$
• D. $x<z<y$

Answer 1

Answer: (A)

$x<y<z$

$x=\tan 15=\tan (45-30)=\frac{\tan 45-\tan 30}{1+\tan 45\tan 30}$

$=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}=\frac{\sqrt{3}-1}{\sqrt{3}+1}=\frac{\sqrt{3}-1}{\sqrt{3}+1}\cdot \frac{\sqrt{3-1}}{\sqrt{3}-1}$

$=\frac{(\sqrt{3}-1{)}^2}{2}=\frac{3+1-2\sqrt{3}}{3}=2-\sqrt{3}$

$y=cosec75=\frac{1}{\sin 75}=\frac{1}{\sin (40+30)}$

$=\frac{1}{\sin 45\cos 30+\cos 45\sin 30}=\frac{1}{\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\cdot \frac{1}{2}}$

$=\frac{2\sqrt{2}}{\sqrt{3}+1}=\frac{2\sqrt{2}(\sqrt{3}-1)}{2}=\sqrt{6}-\sqrt{2}$

$z=4\sin 18=4\left(\frac{\sqrt{5}-1}{4}\right)=\sqrt{5}-1$

Therefore $x<y<z$.