Find the value of $4\sin 36^\circ \cos 18^\circ$ without using their surd values

Question

Without using the surd value for $\sin 36^\circ$ or $\cos 18^\circ$, prove that $$ 4 \sin 36^\circ \cos 18^\circ = \sqrt{5} $$

My attempt:

$ 4\sin36^\circ \cos18^\circ \!\cdot\!\dfrac{\sin18^\circ}{\sin18^\circ}=$

$=\dfrac{2\sin^2 36^\circ}{\sin18^\circ}=$

$=\dfrac{1-\cos72^\circ}{\sin18^\circ}=$

$=\dfrac{1-\sin18^\circ}{\sin18^\circ}=$

$=\dfrac{1}{\sin18^\circ}-1\,.$

And what have you tired? You're expected to show your attempt here. — Sean Roberson, Jul 02 '23 at 04:12
It would be useful if you show how you got to your answer; i.e. the steps in your working — Tony Mathew, Jul 02 '23 at 04:27
Possible hint: try to expand $\sin 5x$ or $\cos 5x$ as a polynomial in $\sin x$ or $\cos x$, and use something about the zeros of that polynomial. — Sean Roberson, Jul 02 '23 at 05:05
I would try this post here for guidance...https://math.stackexchange.com/questions/1524482/proving-the-closed-form-of-sin48-circ (there are methods for finding $\sin 18°$ in this post — bjcolby15, Jul 02 '23 at 14:33

score 2 · Accepted Answer · answered Jul 02 '23 at 19:12

First of all, it results that

$\begin{align}4\cos^2\!18\unicode{176}&=\dfrac{4\cos^3\!18\unicode{176}-3\cos18\unicode{176}}{\cos18\unicode{176}}+3=\dfrac{\cos54\unicode{176}}{\cos18\unicode{176}}+3=\\[3pt]&=\dfrac{\sin36\unicode{176}}{\cos18\unicode{176}}+3=\dfrac{2\sin18\unicode{176}\cos18\unicode{176}}{\cos18\unicode{176}}+3=\\[3pt]&=2\sin18\unicode{176}+3\;\;,\quad\color{blue}{(1)}\end{align}$

$4\sin^2\!18\unicode{176}=1-2\sin18\unicode{176}\,.\;\;\quad\color{blue}{(2)}$

Moreover ,

$\begin{align}4\sin36\unicode{176}\cos18\unicode{176}&=8\sin18\unicode{176}\cos^2\!18\unicode{176}=\\[3pt]&\overset{\color{brown}{(1)}}{=}2\sin18\unicode{176}\big(2\sin18\unicode{176}\!+3\big)=\\[3pt]&=4\sin^2\!18\unicode{176}\!+6\sin18\unicode{176}\overset{\color{brown}{(2)}}{=}4\sin18\unicode{176}+1=\\[3pt]&=\sqrt{\big(4\sin18\unicode{176}\!+1\big)^{\!2}}=\\[3pt]&=\sqrt{16\sin^2\!18\unicode{176}+8\sin18\unicode{176}+1}=\\[3pt]&\overset{\color{brown}{(2)}}{=}\sqrt{4-8\sin18\unicode{176}+8\sin18\unicode{176}+1}=\\[3pt]&=\sqrt5\;.\end{align}$

loved your solution thank you – shiv jindal Jul 03 '23 at 13:51 — shiv jindal, Jul 03 '23 at 13:51
@shivjindal, you are welcome ! – Angelo Jul 03 '23 at 14:32 — Angelo, Jul 03 '23 at 14:32

Find the value of $4\sin 36^\circ \cos 18^\circ$ without using their surd values

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