cosec 18 ° is a root of the equation:

We know that 

$\sin 18°=\frac{\sqrt{5}-1}{4}$

$cosec18°=\frac{1}{\sin 18°} = \frac{4}{\sqrt{5}-1}$

Rationalizing denominator

$=\frac{4}{\sqrt{5}-1}\times \frac{\sqrt{5}+1}{\sqrt{5}+1}$

$=\frac{4\left(\sqrt{5}+1\right)}{4}$

$=\sqrt{5}+1$

Hence equation $x=\sqrt{5}+1$

$\Rightarrow (x-1{)}^2=5$

$\Rightarrow x^2-2x+1 = 5$
$\Rightarrow x^2-2x-4=0$