The quadratic equation whose roots are ${\sin }^2{18}^o$ and cos²36° is

Since, sin²18° and cos²36° are the roots of a quadratic equation
$\therefore$ Sum of roots = sin²18° + cos²36°
$={\left(\frac{\sqrt{5}-1}{4}\right)}^2+{\left(\frac{\sqrt{5}+1}{4}\right)}^2$ 
$=\frac{5+1-2\sqrt{5}}{16}+\frac{5+1+2\sqrt{5}}{16}$ 
$=\frac{12}{16}=\frac{3}{4}$ 
And product of roots = sin²18°cos²36°
$={\left(\frac{\sqrt{5}-1}{4}\right)}^2{\left(\frac{\sqrt{5}+1}{4}\right)}^2$

$={\left(\frac{5-1}{4\times 4}\right)}^2={\left(\frac{1}{4}\right)}^2=\frac{1}{16}$ 
Required equation whose roots are sin²18° and vis
x² - (sum of roots) x+(product of roots) = 0
$\Rightarrow x^2-\frac{3}{4}x+\frac{1}{16}=0$ 

$\Rightarrow 16x^2-12x+1=0$