Q:

How to derive cosine difference formula?

I was asked on a take home quiz to "Derive the Cosine Difference Formula". I've looked for an hour but I've gotten different results this is kind of my last chance. 

A:

And here is a
totally inappropriate way.
$e^{i(a-b)}
= \cos(a-b)+i\sin(a-b)
$
and
$\begin{array}\\
e^{i(a-b)}
&=e^{i(a)}e^{i(-b)}\\
&=(\cos(a)+i\sin(a))(\cos(-b)+i\sin(-b))\\
&=(\cos(a)+i\sin(a))(-i\sin(b))\\
&=\cos(a)\cos(b)+\sin(a)\sin(b)+i(\sin(a)\cos(b)-\cos(a)\sin(b))\\
\end{array}
$
Equating real and imaginary parts,
$\cos(a-b)
=\cos(a)\cos(b)+\sin(a)\sin(b)
$
and
$\sin(a-b)
=\sin(a)\cos(b)-\cos(a)\sin(b)
$.
As is often the case,
there is absolutely
nothing original here.

