Question

Obtain the value of $\frac{\cos {45}^{\circ }}{\sec {30}^{\circ }+\text{cosec}{30}^{\circ }}$

Answer 1

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

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

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

$\Rightarrow$ $\frac{\sqrt{3}}{\sqrt{2}(2+2\sqrt{3})}$

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

$\Rightarrow$ $\frac{\sqrt{3}(2\sqrt{6}-2\sqrt{2})}{(2\sqrt{6}+2\sqrt{2})(2\sqrt{6}-2\sqrt{2})}$

$\Rightarrow$ $\frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{(2\sqrt{6}{)}^2-(2\sqrt{2}{)}^2}$ Using, $(a+b)(a-b)=a^2-b^2$

$\Rightarrow$ $\frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{24-8}$

$\Rightarrow$ $\frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{16}$

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

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