A Question about a trigonometric identity

I am new to Maxima, never having used this mailing list before, and I 
wonder
if anyone can help clarify this small problem I have?.

eq1: R = (3^(3/2)*sin(z)-3*cos(z))/4;
trigrat(%);
print("Maxima fails to simplify the expression. But...");
subst(z1+%pi*2/3,z,eq1);
trigrat(%);
subst(z-%pi*2/3,z1,%);
print("Maxima now shows us that eq1 was in fact just a phase shift");
print("How can I get Maxima to show this without the two subst 
operations?");

My problem of course, is that I had to guess that the expression was a 
simple
phase shift, had I not guessed this, I would never have got there...

Thanks for any help
Martin M

By the way - thank you all for supporting Maxima. I had the pleasure of
working with Macsyma on the Symbolic XL1200 -- I have the original
Symbolics Macsyma documentation someplace. It is wonderful to have
access to Maxima as an open source project. Thank you.