## Saturday, 9 May 2009

### Trig identities

I always have trouble remembering these, here's a simple derivation using Euler's formula:

\dvipng\gammacorrection{1.2}\begin{align*}e^{i(A+B)} &= \cos (A+B) + i \sin (A+B)\\ e^{iA}e^{iB} &= (\cos A + i \sin A)(\cos B + i \sin B) \\ &= \cos A \cos B + i^2 \sin A \sin B \\ & \qquad + \cos A i \sin B + i \sin A \cos B. \end{align*}

Note that the imaginary and real parts are separate so the line $\dvipng\gammacorrection{1.2} \cos A \cos B + i^2 \sin A \sin B$, which is real, corresponds to $\dvipng\gammacorrection{1.2} \cos (A+B)$ on the top line.