Question

Evaluate the following:
(iii) $\frac{\text{cos}{45}^{\circ }}{(\text{sec}{30}^{\circ }+\text{cosec}{30}^{\circ })}$

Answer 1

(iii) $\frac{\text{cos}{45}^{\circ }}{(\text{sec}{30}^{\circ }+\text{cosec}{30}^{\circ })}$
$=\frac{\frac{1}{\sqrt{2}}}{(\frac{2}{\sqrt{3}}+2)}=\frac{\frac{1}{\sqrt{2}}}{\frac{(2+2\sqrt{3})}{\sqrt{3}}}$
$=\frac{\sqrt{3}}{\sqrt{2}\times (2+2\sqrt{3})}=\frac{\sqrt{3}}{(2\sqrt{2}+2\sqrt{6})}$
$=\frac{\sqrt{3}\times (2\sqrt{6}-2\sqrt{2})}{(2\sqrt{6}+2\sqrt{2})\times (2\sqrt{6}-2\sqrt{2})}$
$=\frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{(2\sqrt{6}{)}^2-(2\sqrt{2}{)}^2}$
$=\frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{(24-8)}=\frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{16}$
$=\frac{\sqrt{3}(\sqrt{6}-\sqrt{2})}{8}=\frac{(\sqrt{18}-\sqrt{6})}{8}=\frac{(3\sqrt{2}-\sqrt{6})}{8}$