Finding exact values for trigonometric functions

(d) Hence find the exact values of $\tan \left (\frac{k}{16} \pi \right )$ for $k=1,5,9 \text{ and } 16$.
Any ideas for part d?

If you let $t=\tan(\theta)$ then you get the quartic when $\tan(4\theta)=1$.

So find the solutions of that.

For d) Find what the sum of the squares of the roots is. Use $\Sigma \alpha^2 = (\Sigma \alpha)^2 - 2\Sigma \alpha \beta$

You might also need to convert some of the roots from c) so you answer d)

Hope that helps