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This resource is from Underground Mathematics.

 

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?

 

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