There is a similar, and in my view slightly more elegant, way of deriving the compound angle formulas for sine and cosine.

I'd attach a picture but I can't see any way to add attachments to this so I'll post the tikz code instead:

\usepackage{tikz-euclide}
\usetkzobj{all}

\begin{document}
\begin{tikzpicture}
\tkzDefPoints{0/0/A, 6/0/B, 6/4/C, 0/4/D}
\tkzDefPoint(70:5){E}
\tkzInterLL(A,E)(C,D) \tkzGetPoint{E}
\tkzDefShiftPoint[E](-20:6){F}
\tkzInterLL(E,F)(B,C) \tkzGetPoint{F}
\tkzDrawPolygon(A,B,C,D)
\tkzDrawPolygon(A,E,F)
\tkzLabelSegment[above](A,F){1}
\tkzMarkRightAngle[size=0.2](A,E,F)
\tkzMarkAngle[size=0.5](F,A,E)
\tkzLabelAngle[pos=0.8](F,A,E){$\beta$}
\tkzMarkAngle[size=0.7](E,A,D)
\tkzLabelAngle[pos=1.1](E,A,D){$\alpha$}
\end{tikzpicture}
\end{document}

Credit for the idea to Phillips Exeter Academy's maths department.