<s>[INST] Solve $\sin(\frac{\pi}{5})$ analytically How do I solve $\sin(\frac{\pi}{5})$ analytically? I looked up here and in the first step I have to 

Show that: $\cos \left(\frac{\pi }{5}\right)-\sin
 \left(\frac{\pi }{10}\right)=\frac{1}{2}$

My question is, why do I have to do that?
 [/INST] By repeated application of angle sum formulas we may get,
$$\sin (5x)=\sin^5 x+5 \cos^4 x\sin x-10 \sin^3 x \cos^2 x$$
Let $x=\frac{\pi}{5}$ and let $\sin (\frac{\pi}{5})=u$ then we have,
$$0=u^5+5(1-u^2)^2 u-10(1-u^2)u^3$$
It is safe to say $\frac{\sqrt{2}}{2}>u>0$. So that we may divide by $u$ to get.
$$0=u^4+5(1-u^2)^2-10(1-u^2)u^2$$
$$0=16u^4-20u^2+5$$
By solving this for $u^2$ first and then $u$ you get the only root in $(0, \frac{\sqrt{2}}{2})$ to be,
$$u=\frac{1}{2} \sqrt{\frac{5}{2}-\frac{\sqrt{5}}{2}}$$
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