I showed this first result and after that for $x^4-10x^2+5=0$, I solved for $\tan 5\theta=0$, I understand all this , but then I get $\theta=\pi/5$. I know I have to multiply by $n$ to get 5 solutions, but I have no idea why I even do that, and in the examiner report they tell that some justification is needed as to why $n$ is included rather than just stating it, I don't know what to write for that. What should I write after the step $\tan5\theta=0$, $\theta=\pi/5$, what is the step after this before saying it's $\tan(n\pi/5)$?
Then in the next part they say product of the roots, but the product of the roots is $$\tan (\pi/5)\tan(2\pi/5)\tan(3\pi/5)\tan(4\pi/5)$$ But they have said the roots are $$\pm\tan (\pi/5), \pm\tan(2\pi/5)$$ Please help me in this too.
Since $$t^5-10t^3+5t=t(t^4-10t^2+5)=0$$ we know that if $x=t=\tan\theta$ is a root of $x^4-10x^2+5$, then $\tan5\theta=0$ but $\tan\theta\neq0$. The solutions of $\tan(5\theta)=0$ for $\theta\in[0,2\pi)$ are $$\theta=0,\frac{n\pi}{5}$$ for $n=1,2,\ldots,9$. But since we care only about distinct values of $x$, there is a lot of redundancy here. If $\theta=0$ or $n=5$ (so $\theta=\pi$), then $\tan\theta=0$, violating the above assumption that $\tan\theta\neq 0$. Moreover, $$\tan \frac{6\pi}{5}=\tan\frac{\pi}{5}$$ $$\tan \frac{7\pi}{5}=\tan\frac{2\pi}{5}$$ $$\tan \frac{8\pi}{5}=\tan\frac{3\pi}{5}$$ $$\tan \frac{9\pi}{5}=\tan\frac{4\pi}{5}$$ So we only get four distinct roots, $x=\tan\frac{n\pi}{5}$ for $n=1,2,3,4$. You could also have chosen $n=6,7,8,9$, or $n=1,7,3,4$, etc... The important point is to select four values of $n$ that give you the four different roots of $x^4-10x^2+5$.