Use de Moivre's theorem to prove that $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ State the exact values of \(\theta\), between 0 and \(\pi\), that satisfy \(\tan 3 \theta = 1\). Express each root of the equation \(t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0\) in the form \(\tan ( k \pi )\), where \(k\) is a positive rational number. For each of these values of \(k\), find the exact value of \(\tan ( k \pi )\).

Use de Moivre's theorem to prove that

$$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$

State the exact values of $\theta$, between 0 and $\pi$, that satisfy $\tan 3 \theta = 1$.

Express each root of the equation $t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0$ in the form $\tan ( k \pi )$, where $k$ is a positive rational number.

For each of these values of $k$, find the exact value of $\tan ( k \pi )$.

\hfill \mbox{\textit{CAIE FP1 2011 Q11 EITHER}}