Question

Simplify: $\cos 45°÷(\sec 30°+\cos ec30°)$

Answer 1

We know that,

$\cos {45}^{\circ }=\frac{1}{\sqrt{2}}$

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

$\cos ec{30}^{\circ }=2$

$\therefore \cos 45°÷(\sec 30°+\cos ec30°)=\frac{1}{\sqrt{2}}÷(\frac{2}{\sqrt{3}}+2)$

$=\frac{1}{\sqrt{2}}÷\left(\frac{2+2\sqrt{3}}{\sqrt{3}}\right)$

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

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

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

$=\frac{2\sqrt{6}-6\sqrt{2}}{8-24}$

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