<s>[INST] Calculate $\cos(\frac{2\pi}{5})$ in terms of $\sin(\frac{\pi}{5})$ I intend to calculate $\cos\left(\frac{2\pi}{5}\right)$ via the formula 
$$\sin\left(\frac{\pi}{5}\right)=\sqrt{\frac{1-\cos\left(\frac{2\pi}{5}\right)}{2}}$$
This could be expressed as:
$\sqrt{\frac{5-\sqrt{5}}{8}}=\sqrt{\frac{1-\cos\left(\frac{2\pi}{5}\right)}{2}}$. Squaring both sides and manipulate a bit, this will give $\cos\left(\frac{2\pi}{5}\right)=-\frac{1-\sqrt{5}}{4}$. However, the true result is $\frac{\sqrt{5}-1}{4}$. What is wrong with my derivation?
 [/INST] Note that
$$-\frac{1-\sqrt{5}}{4}=(-1)\cdot\frac{1-\sqrt{5}}{4}=\frac{(-1)\cdot1-(-1)\cdot(\sqrt{5})}{4}=\frac{\sqrt{5}-1}{4}$$
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