Question

Prove that
$\cos {36}^{\circ }=\frac{\sqrt{5}+1}{4}$

Answer 1

we use the formula:

$\cos 5\theta =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta$

Put $\theta ={36}^{\circ }$ we get,

$\cos 180=16{\cos }^{5}36-20{\cos }^{3}36+5\cos 36$

substitute $\cos 36=x$

$-1=16x^5-20x^3+5x$

Solving the above equation using Newton Raphson method, we get,

$x=-1,\frac{1\pm \sqrt{5}}{4}$

$\therefore \cos {36}^{\circ }=\frac{1+\sqrt{5}}{4}$