Question

Simplify:

$\frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+\mathrm{cosec}{30}^{\circ }}$

Answer 1

Given

$\frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+cosec{30}^{\circ }}$
$=\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+\frac{2}{1}}$
$=\frac{1}{\sqrt{2}}\times \frac{1}{\frac{2+2\sqrt{3}}{\sqrt{3}}}$
$=\frac{\sqrt{3}}{2\sqrt{2}(\sqrt{3}+1)}$

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

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

$=\frac{\sqrt{3}(\sqrt{3}-1)}{4\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}$

$=\frac{\sqrt{6}(\sqrt{3}-1)}{8}$

$=\frac{\sqrt{18}-\sqrt{6}}{8}$

$=\frac{3\sqrt{2}-\sqrt{6}}{8}$