Question

The value of $\frac{1}{\cos {285}^o}+\frac{1}{\sqrt{3}\sin {255}^o}$, is

• A. $2\sqrt{2}$
• B. $\frac{4\sqrt{2}}{\sqrt{3}}$
• C. $\frac{2\sqrt{2}}{3}$
• D. $\sqrt{3}-\sqrt{2}$

Answer 1

The correct option is B $\frac{4\sqrt{2}}{\sqrt{3}}$

We need to find value of $\frac{1}{\cos {285}^o}+\frac{1}{\sqrt{3}\sin {255}^o}$

It can be written as $\frac{1}{\cos (270+15{)}^o}+\frac{1}{\sqrt{3}\sin (270-15{)}^o}$

$=\frac{1}{\sin {15}^o}+\frac{1}{\sqrt{3}(-\cos {15}^o)}=\frac{\sqrt{3}\cos {15}^o-\sin {15}^o}{\sqrt{3}\sin {15}^o\cos {15}^o}$

$=\frac{2\left(\cos \right({30}^o+{15}^o\left)\right)}{\frac{\sqrt{3}}{2}\times \sin {30}^o}$

$=\frac{2\cos {45}^o}{\left(\sqrt{3}/4\right)}=\frac{8}{\sqrt{3}}\times \frac{1}{\sqrt{2}}=\frac{4\sqrt{2}}{\sqrt{3}}$