Question

Find the value of the following:
$\frac{\cos {45}^o}{\sec {30}^o+\csc {30}^o}$

Answer 1

We know that the values of the trignometric functions $\csc {30}^0=2$, $\sec {30}^0=\frac{2}{\sqrt{3}}$ and $\cos {45}^0=\frac{1}{\sqrt{2}}$, therefore, the value of $\frac{\cos {45}^0}{\sec {30}^0+\csc {30}^0}$ is:

$\frac{\cos {45}^0}{\sec {30}^0+\csc {30}^0}=\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+2}=\frac{\frac{1}{\sqrt{2}}}{\frac{2+2\sqrt{3}}{\sqrt{3}}}=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2+2\sqrt{3}}=\frac{\sqrt{3}}{\sqrt{2}(2+2\sqrt{3})}=\frac{\sqrt{3}}{2\sqrt{2}+2\sqrt{6}}$

Hence, $\frac{\cos {45}^0}{\sec {30}^0+\csc {30}^0}=\frac{\sqrt{3}}{2\sqrt{2}+2\sqrt{6}}$.