If we write \(t = \tan \theta\), then the following equations are true.

\[\begin{align*} \tan 2\theta &= \frac{2t}{1-t^2}, \\ \sin 2\theta &= \frac{2t}{1+t^2}, \\ \cos 2\theta &= \frac{1-t^2}{1+t^2}. \end{align*}\]Can you use this diagram to obtain these formulae?

For what range of values of \(\theta\) does this argument work?