TITLE: Intuition of Addition Formula for Sine and Cosine
QUESTION [27 upvotes]: The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the proofs using acute angles which can be seen here http://en.wikibooks.org/wiki/Trigonometry/Addition_Formula_for_Cosines but pretty sure those who have taken trig know what I'm talking about. So I know how to derive and prove both of the two-angle functions using the acute angles, but what I am completely confused about is where those triangles came from. So for proving the two-angle cosine function, we look at two acute angles, $A$ and $B$, where $A+B<90$ and keep on expanding. So my question is, where did those two triangles come from and what is the intuition behind having two acute triangles on top of each other?

REPLY [10 votes]: My physics professor was from Germany, and he said High School students were taught to just rederive trig identities any time they needed them using complex "algebra". That is, you can write out: 
\begin{align}
\cos(x+y)+i\sin(x+y)
 & = e^{i(x+y)}= e^{ix}e^{iy}\\
 & = (\cos(x)+i\sin(x))(\cos(y)+i\sin(y))\\
 & = \cos(x)\cos(y)-\sin(x)\sin(y)+i(\cos(x)\sin(y)+\sin(x)\cos(y))
\end{align}
pretty quickly, and get the desired identity by considering real or imaginary part of equation. For instance, try this for $e^{i2x}=e^{ix}e^{ix}$.