<s>[INST] Calculate $\tan^2{\frac{\pi}{5}}+\tan^2{\frac{2\pi}{5}}$ without a calculator The question is to find the exact value of:
$$\tan^2{\left(\frac{\pi}{5}\right)}+\tan^2{\left(\frac{2\pi}{5}\right)}$$
without using a calculator.
I know that it is possible to find the exact values of $\tan{\left(\frac{\pi}{5}\right)}$ and $\tan{\left(\frac{2\pi}{5}\right)}$ to find that the answer is $10$. However, I want to know whether there is a faster way that does not involve calculating those values.
So far, I have this: let $a=\tan{\left(\frac{\pi}{5}\right)}$ and $b=\tan{\left(\frac{2\pi}{5}\right)}$; then, $b=\frac{2a}{1-a^2}$ and $a=-\frac{2b}{1-b^2}$, so multiplying the two and simplifying gives:
$$a^2+b^2={\left(ab\right)}^2+5$$
Any ideas? Thanks!
 [/INST] Inspired/ Trying to make sense of Jaideep's Answer ...
\begin{eqnarray*}
\tan(5 \theta) =\frac{5 \tan(\theta)-10 \tan^3(\theta)+\tan^5(\theta)}{1-10\tan^2(\theta) +5\tan^4(\theta)}
\end{eqnarray*}
Let $\theta=\frac{\pi}{5}$ and $x=tan(\frac{\pi}{5})$. We have the quintic
\begin{eqnarray*}
5x-10x^3+x^5=0
\end{eqnarray*}
This has roots $tan(\frac{\pi}{5}),tan(\frac{2\pi}{5}),tan(\frac{3\pi}{5}),tan(\frac{4\pi}{5}),tan(\frac{5\pi}{5})$.
Note that $tan(\frac{3\pi}{5})=-tan(\frac{2\pi}{5})$ , $tan(\frac{4\pi}{5})=-tan(\frac{\pi}{5}) $ and $tan(\frac{5\pi}{5})=0$. 
Now square this quintic & substitute $y=x^2$ and we have 
\begin{eqnarray*}
25y-100y^2+110y^3-\color{red}{20}y^4+y^5=0
\end{eqnarray*}
The sum of the roots of this polynomial gives $2 (tan^2(\frac{\pi}{5})+tan^2(\frac{2\pi}{5}))=20$.
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