vin,
you can do this:
Simplify[rule1 == 0, {a > 0, b > 0}]
([Pi] (-a b (b Bmn Dqx m + a Cmn Dqy n) +
Amn (b^2 m^2 (Dqx + Nx) + a^2 n^2 (Dqy + Ny)) [Pi]) -
a^2 Amn b^2 h [Rho] [Omega]mn^2) Sin[(m [Pi] x)/a] Sin[(
n [Pi] y)/b] == 0
or
Assuming[{a > 0, b > 0}, Simplify[rule1 == 0]]
Note that the Sin terms cannot be divided out because they can be zero. If you are certain they can never be zero:
Assuming[{a > 0, b > 0, Sin[(m \[Pi] x)/a] != 0,
Sin[(n \[Pi] y)/b] != 0}, Simplify[rule1 == 0]]
The other way to look at this is your equation is solved if either of the sin terms are zero or (by assuming they are not zero) the resulting equation is satisfied.
I hope this helped.
Regards,
Neil
