We know that, & 2 =1-2 ^2 & 36^ = 2 (18^ ) & =1-2 ^2 18^ & =1-2 ( 5-1 4 )^2 & = 8-(5+1-2 5) 8 & = 8-(6-2 5) 8 & = 2+2 5 8 & 36^ = 5+1 4

36^ = 5+1 4

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Question

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

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Answer

We know that,
$\begin{aligned}
& \cos 2 \theta=1-2 \sin ^2 \theta \\
& \cos 36^{\circ}=\cos 2\left(18^{\circ}\right) \\
& =1-2 \sin ^2 18^{\circ} \\
& =1-2\left(\frac{\sqrt{5}-1}{4}\right)^2 \\
& =\frac{8-(5+1-2 \sqrt{5})}{8} \\
& =\frac{8-(6-2 \sqrt{5})}{8} \\
& =\frac{2+2 \sqrt{5}}{8} \\
& \therefore \cos 36^{\circ}=\frac{\sqrt{5}+1}{4}
\end{aligned}$

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