<s>[INST] Various methods to find value of $\sin 18^\circ$ To find value of $\sin 18^\circ$. Now my textbook gives a proof in which it takes $\theta=18^\circ$ and then multiply it out by 5 and write again as sum of $2\theta+3\theta$ and then taking sin on both sides forms a quadratic equation.
Are there any other elegant ways to find its value?
Thanks
 [/INST] Let $x = \sin(18^{\circ})\implies x^2 = \dfrac{1-y}{2}$, whereas $y = \cos(36^{\circ})$. And square again $y^2 = \dfrac{1+\cos(72^{\circ})}{2}= \dfrac{1+x}{2}\implies 2x^2+y =  1 = 2y^2-x\implies 2(x^2-y^2) + (x+y) = 0\implies (x+y)(2x-2y+1) = 0\implies 2x= 2y-1\implies 4x^2=2-2y = 1-2x\implies 4x^2+2x-1=0$. You can now solve for $x$ to get the answer in terms of radical which is the expected form of the answer.
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