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FWIW :
sage: th = var('th')
sage: f = function('f')(th)
sage: omega = function('omega')(th)
sage: eq = 2*pi*cos(th)*sin(th)*diff(omega,th,th)
sage: w0 = SR.wild(0)
sage: eq.subs([w0*sin(th)*cos(th)==w0*f])
2*pi*f(th)*diff(omega(th), th, th)
The use of a wildcard allows for partial substitution...
As for Maxima :
sage: f.maxima_methods().ratsubst(cos(th)*sin(th),eq)
2*pi*f(th)*diff(omega(th), th, th)
Also :
sage: maxima_calculus.lratsubst([cos(th)*sin(th)==f],eq)._sage_()
2*pi*f(th)*diff(omega(th), th, th)
HTH,
