Question

Evaluate $\frac{\cos 45°}{(sec30°+\cos ec30°)}$

Answer 1

Step 1

Recall the values of required angles from the trigonometry values table

So

$$\begin{gathered} C\mathrm{os}45°=\frac{1}{\sqrt{2}} \\ Sec30°=\frac{2}{\sqrt{3}} \\ C\mathrm{os}ec30°=2 \end{gathered}$$

Step 2

Put values in the given expression and simplify by taking LCM

$\begin{array}{rcl}& \Rightarrow & \frac{\cos 45˚}{(sec30°+\cos ec30°)}=\frac{{\displaystyle \frac{1}{\sqrt{2}}}}{{\displaystyle \frac{2}{\sqrt{3}}}+{\displaystyle \frac{2}{1}}}=\frac{{\displaystyle \frac{1}{\sqrt{2}}}}{{\displaystyle \frac{2+2\sqrt{3}}{\sqrt{3}}}}=\frac{\sqrt{3}}{\sqrt{2}(2+2\sqrt{3})}\\ & =& \frac{\sqrt{3}}{2\sqrt{2}(1+\sqrt{3})}\end{array}$

Step 3

Rationalize the terms

$\begin{array}{rcl}& \Rightarrow & \frac{\sqrt{3}}{2\sqrt{2}(\sqrt{3}+1)}=\frac{\sqrt{3}}{2\sqrt{2}(\sqrt{3}+1)}\times \frac{(\sqrt{3}-1)}{(\sqrt{3}-1)}\\ & =& \frac{\sqrt{3}(\sqrt{3}-1)}{2\sqrt{2}(3-1)}=\frac{(3-\sqrt{3})}{2\sqrt{2}\times 2}\\ & =& \frac{(3-\sqrt{3})}{4\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{3\sqrt{2}-\sqrt{6}}{8}\end{array}$

Hence we get $\frac{3\sqrt{2}-\sqrt{6}}{8}$ after simplification.