Question

Evaluate:$\frac{\cos {45}^0}{(\sec {30}^0+\mathrm{cosec}{30}^0)}$

Answer 1

Solve by using values of trigonometric function:

$\frac{\cos {45}^0}{(\sec {30}^0+\mathrm{cosec}{30}^0)}$ $=\frac{\frac{1}{\surd 2}}{{\displaystyle \frac{2}{\surd 3}}+2}$

$=\frac{\frac{1}{\surd 2}}{{\displaystyle \frac{2+2\surd 3}{\surd 3}}}$

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

$=\frac{\surd 3}{2\surd 2(1+1\surd 3)}$

$=\frac{\surd 3}{2\surd 2(1+\surd 3)}\times \frac{(1-\surd 3)}{(1-\surd 3)}$

$=\frac{\surd 3(1-\surd 3)}{2\surd 2{\left[\right(1)}^2-{(\surd 3)}^2]}$

$=\frac{\surd 3(1-\surd 3)}{2\surd 2(1-3)}$

$=\frac{\surd 3(1-\surd 3)}{2\surd 2(-2)}$

$=\frac{\surd 3(1-\surd 3)}{-4\surd 2}$

$=\frac{\surd 3-3}{-4\surd 2}$

$=\frac{3-\surd 3}{4\surd 2}\times \frac{\surd 2}{\surd 2}$

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

Hence, value found is $\frac{3\surd 2-\surd 6}{8}$.