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?
![Right angled triangle containing the angle theta and with base length 2. A line is drawn from the top corner of the right anlged triangle to the base, forming an isoceles triangle with the equal angles being theta](/trigonometry-compound-angles/t-for-tan/images/diagram1.png)
For what range of values of \(\theta\) does this argument work?