trigexpand(), trigsimp() and an oscillatory solution of a 2nd order linear differential equation

I was working on a differential equations homework problem and it took me a few tries to remember how to simplify a trig expression.  So, I thought I’d put the results here to remind myself and others about trigexpand().

The second order linear equation is

$y''+y'+y=0 \;\;\;\;\;\; y(0)=1, y'(0)=0$

Now I asked my students to use the identities

$A= \sqrt{C_1^2 + C_2^2}$  and $\tan \phi=C_2/C_1$

to write the equation equivalently as

$y=A e^{-x/2} \cos \left( \frac{\sqrt{3}}{2} x -\phi \right)$

Finally, I wanted to use Maxima to show these two expressions are equivalent.  I reflexively used trigsimp() on the difference between the two expression   (the name says it all, am I right?)  but the result was complicated in a way that I couldn’t recover…certainly not zero as I hoped.  A few minutes of thought and I remembered trigexpand() which, followed by a call to trigsimp(), gives the proof: