ALSO -- Fu, et al., have already thought a lot about trig simplification 
and if you just run fu(expr) you may get what you are looking for:

>>> ok = fu(P.expand()); print filldedent(ok)

D**2 + 2*D*F*sin(x) + 2*D*G*cos(x) + 2*D*H*sin(2*x) + 2*D*J*cos(2*x) +
F**2*(-cos(2*x)/2 + 1/2) + F*G*sin(2*x) + F*H*(cos(x) - cos(3*x)) +
F*J*(-sin(x) + sin(3*x)) + G**2*(cos(2*x)/2 + 1/2) + G*H*(sin(x) +
sin(3*x)) + G*J*sin(4*x)/(2*sin(x)) + H**2*(-cos(4*x)/2 + 1/2) +
H*J*sin(4*x) + J**2*(cos(4*x)/2 + 1/2)

>>> print filldedent(ok.atoms(Pow))

set([D**2, J**2, F**2, 1/sin(x), G**2, H**2])


There are no powers of trig functions except for the sin(x) in the 
denominator.

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