Simplifying $\csc(\frac{5\pi }{12})$

Question

I need to simplify $\csc(\frac{5\pi }{12})$

I got to the point $\frac{2 \sqrt[]{2} }{ \sqrt[]{3}+1}$ but I have to get to $\sqrt[]{6}-\sqrt[]{2}\}$

Could someone guide me the way?

Answer 1

Just follow the suggestion of Zack Ni: $$\frac{2 \sqrt[]{2} }{ \sqrt{3}+1}=\frac{2 \sqrt[]{2} }{ \sqrt{3}+1}\cdot \frac{\sqrt{3}-1}{ \sqrt{3}-1}= \frac{2 \sqrt{6}-2 \sqrt{2} }{ 3-1}=\sqrt{6}-\sqrt{2}.$$ More generally $$\frac{c}{ \sqrt{a}+b}=\frac{c}{ \sqrt{a}+b}\cdot \frac{ \sqrt{a}-b}{ \sqrt{a}-b}= \frac{c\sqrt{a}-cb}{a-b^2}.$$