Question

Evaluate:

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

Answer 1

As, We know that, $\cos {45}^{\circ }=\frac{1}{\sqrt{2}}$

$\sec {30}^{\circ }=\frac{2}{\sqrt{3}}$ and $\cos {30}^{\circ }=\frac{2}{1}$

$\therefore \frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+\csc {30}^{\circ }}$

$=\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+2}$

$=\frac{\sqrt{3}}{2\sqrt{2}+2\sqrt{6}}$

Hence, $\frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+\text{cosec}{30}^{\circ }}=\frac{\sqrt{3}}{2\sqrt{2}+2\sqrt{6}}$