<s>[INST] Finding the value of given integral The given integral is :$$\displaystyle\int_0^1 \dfrac{\sin{\theta}(\cos^2{\theta} -\cos^2{\frac{\pi}{5}})(\cos^2{\theta} -\cos^2{\frac{2\pi}{5}})}{\sin{5\theta}} \, d\theta$$
I tried solving it with some trigonometric identities but comes out it does not work here.
However solution given starts with these equations mentioned down below and later they substituted $z= \cos{\theta}+ i\sin(\theta)$
$$\begin{array}{r}
z^{10}-1=\left(z^2-1\right)\left[z^2-2 \cos \left(\frac{\pi}{5}\right) z+1\right]\left[z^2-2\left(\cos \frac{2 \pi}{5}\right) z+1\right] \\
\quad \times\left(z^2-2 \cos \frac{4 \pi}{5} z+1\right)\left(z^2-2 \cos \frac{6 \pi}{5} z+1\right)
\end{array}$$
$$\begin{aligned}
z^5-\frac{1}{z^5}=\left(z-\frac{1}{z}\right) &\left(z-2 \cos \frac{\pi}{5}+\frac{1}{z}\right)\left(z-2 \cos \frac{2 \pi}{5}+\frac{1}{z}\right) \\
& \times\left(z-2 \cos \frac{4 \pi}{5}+\frac{1}{z}\right)\left(z-2 \cos \frac{6 \pi}{5}+\frac{1}{z}\right)
\end{aligned}$$
Can anyone help me understand how I can derive these equations and use them here in this integral?
Thank you for your help.
 [/INST] First you should simplify the term and then bild the antiderivative, what is really easy after simplifying the term:
$$
\begin{align*}
y &= \int_{0}^{1} \dfrac{\sin{\theta}(\cos^2{\theta} -\cos^2{\frac{\pi}{5}})(\cos^2{\theta} -\cos^2{\frac{2\pi}{5}})}{\sin{5\theta}} ~\operatorname{d}\theta\\
F(\theta) &= \int_{0}^{\theta} \dfrac{\sin{\theta}(\cos^2{\theta} -\cos^2{\frac{\pi}{5}})(\cos^2{\theta} -\cos^2{\frac{2\pi}{5}})}{\sin{5\theta}} ~\operatorname{d}\theta = \int_{0}^{\theta} f(\theta) ~\operatorname{d}\theta\\
f(\theta) &= \dfrac{\sin{\theta}(\cos^2{\theta} -\cos^2{\frac{\pi}{5}})(\cos^2{\theta} -\cos^2{\frac{2\pi}{5}})}{\sin{5\theta}}\\
f(\theta) &= \dfrac{1}{16}\\
F(\theta) &= \int_{0}^{\theta} \dfrac{1}{16} ~\operatorname{d}\theta\\
F(\theta) &= \dfrac{\theta}{16}\\
y &= \int_{0}^{1} \dfrac{\sin{\theta}(\cos^2{\theta} -\cos^2{\frac{\pi}{5}})(\cos^2{\theta} -\cos^2{\frac{2\pi}{5}})}{\sin{5\theta}} ~\operatorname{d}\theta = F(1) - F(0)\\
F(1) - F(0) &= \dfrac{1}{16} - \dfrac{0}{16}\\
F(1) - F(0) &= \dfrac{1}{16} - 0\\
F(1) - F(0) &= \dfrac{1}{16}\\
F(1) - F(0) &= 0.0625\\
\end{align*}
$$
$$ 
\int_{0}^{1} \dfrac{\sin{\theta}(\cos^2{\theta} -\cos^2{\frac{\pi}{5}})(\cos^2{\theta} -\cos^2{\frac{2\pi}{5}})}{\sin{5\theta}} ~\operatorname{d}\theta = 0.0625\\ $$
If you can't simplefy it, you also would be able to integrate it without simplefying it but that's way more complex and unambiguously.
</s>