Question

find the value of cos 45 / sec30 + cosec 30

Answer 1

Cos 45 / Sec 30 + Cosec 30
=(1/root 2)÷(2/root 3) + 2
=(1×root3)÷(2root2)+2
=(root3+4×root2)÷2×root2.

Answer 2

$The \ value \ of \ \frac{\left(\cos 45^{\circ}\right)}{\sec 30^{\circ}+\csc 30^{\circ}} = \frac{3 \sqrt{2} - \sqrt{6}}{8}$

Given:

$\cos 45^{\circ}, \sec 30^{\circ}, \csc 30^{\circ}$

To find:

$\frac{\left(\cos 45^{\circ}\right)}{\sec 30^{\circ}+\csc 30^{\circ}} = ?$

Solution:

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

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

$\csc 30^{\circ} = 2$

$\frac{\left(\cos 45^{\circ}\right)}{\sec 30^{\circ}+\csc 30^{\circ}} = \frac{\frac{1}{\sqrt{2}}}{\left(\frac{2}{\sqrt{3}}+2\right)}$

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

$\Rightarrow \frac{\left(\cos 45^{\circ}\right)}{\sec 30^{\circ}+\csc 30^{\circ}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{(2(1+\sqrt{3}))}$

Rationalizing the denominator,

$\Rightarrow\left(\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}\right) \times\left(\frac{\sqrt{3}}{(2(1 + \sqrt{3}))} \times \frac{1 - \sqrt{3}}{1-\sqrt{3}}\right)$

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

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

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

$\Rightarrow \frac{\left(\cos 45^{\circ}\right)}{\sec 30^{\circ}+\csc 30^{\circ}} = \frac{3 \sqrt{2}-\sqrt{6}}{8} \frac{3 \sqrt{2}-\sqrt{6}}{8}$