Do you know a mnemonic to remember Prosthaphaeresis formulas?

[LCSphysicist]

Homework Statement

Do you know a mnemonic to remember Prosthaphaeresis formulas?

Relevant Equations

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Do you have a mnemoic to renember this four formulas?:

It is pretty easy to derive, but a little boring do the derivation all the time.

 

[archaic]

No, but you can memorize Euler's formula$$e^{ix}=\cos x+i\sin x$$from which you can get, through exploiting the fact that $\sin(-x)=-\sin x$, and $\cos(-x)=\cos x$$$\begin{cases}
\cos x=\frac{e^{ix}+e^{-ix}}{2}\\
\sin x=\frac{e^{ix}-e^{-ix}}{2i}
\end{cases}$$Let's try $2\sin a\sin b$:
$$\begin{align*}
2\sin a\sin b&=2\frac{e^{ia}-e^{-ia}}{2i}\frac{e^{ib}-e^{-ib}}{2i}\\
&=\frac{e^{ia}e^{ib}-e^{ia}e^{-ib}-e^{-ia}e^{ib}+e^{-ia}e^{-ib}}{2(-1)}\\
&=-\frac{e^{i(a+b)}-e^{i(a-b)}-e^{-i(a-b)}+e^{-i(a+b)}}{2}\\
&=-\frac{\left(e^{i(a+b)}+e^{-i(a+b)}\right)-\left(e^{i(a-b)}+e^{-i(a-b)}\right)}{2}\\
&=\frac{e^{i(a-b)}+e^{-i(a-b)}}{2}-\frac{e^{i(a+b)}+e^{-i(a+b)}}{2}\\
&=\cos(a-b)-\cos(a+b)
\end{align*}$$hooray!