I think it is more clear than it displays with all the parameters:

The 1st means: p = constant1 * q where this constant1 depends on (... what you easily see ...).

The 2nd should be read as constant2 * q = 0 (one has to know that Maple abbreviates a bit here), where this constant2 depends on your parameters).

So you have to discuss for which parameters that constant2 equals 0, if you want solutions beyond q=0=p.

You may post / upload one or two examples for which you have problems.

This equation can be solved 'directly', exact.

For small values (say below 1E-4) the value is approximated by
1.23107661228290 - 0.553095709523571*log(Vzero)

I always try to get rid of denominators (just multiply [and check finally]), here giving the linear system - for which fsolve works without pain.

Yes, for example used by Bourbaki (btw deg <= ... ) for the degree

Guessing you have in mind some compiled version (k=352000000, Digits=18 for presentations of IEEE doubles <-> decimals ):

2*k*Pi 'is' 1159557927732361/2251799813685248 * pow(2,32) and sinus of it (with higher precision) is -0.811500562932876925e-8 or -2452609957903403/4503599627370496 * pow(2,-26)

avoiding any "equation labels"

Testing for "%add.mw" (Win 7, Maple 32 bit): I do not have Maple 2018, but for Maple 2017 it fails  and for Maple 18 and Maple 2016 it works. For classical sheets *.wms it works in all cases. A crosscheck with the "DOS prompt" shows no unexpected characters (as it sometimes happens for files from an Apple machine)

With some modification "split and symmetry" works to show it is zero:

# using symmetry in Pi/2
2*Int(cos(2*x)/(1+2*sin(3*x)^2), x = 0 .. Pi/2);
Change(%, 2*x=xx,xx): subs(xx=x, %): combine(%);
 
# split and reduce to common range
Split(%, [Pi/3, 2*Pi/3]);
op(1, %) + Change(op(2, %), x = Pi/3 + w) + Change(op(3, %), x = 2*Pi/3 + w);
combine(%): expand(%): combine(%);
 
# which is zero by symmetry of the integrand
op(1, %):
'eval(%, w = Pi/6 - w) = -eval(%, w = Pi/6 + w)';
expand(%): is(%);

@digerdiga what are your ranges for t?

@vv for correcting me

The integral is (bounded and oscillating) * 1/k, so it does not converge

@vv 

I meant the limit over Gamma*exp (after partial integration w.r.t. Gamma(a,s)), 0<a,0<b.

That works:

limit(-1/b*exp(-b*s) * GAMMA(a,s),s = infinity);
eval(%, limit=MultiSeries:-limit);

                                 exp(-b s) GAMMA(a, s)
                     lim       - ---------------------
                s -> infinity              b

                                  0

While that does not work:

MultiSeries:-limit(exp(-b*s) * GAMMA(a,s),s = 0,right);

@vv 

it seems that "limit" does not always know or use https://fr.wikipedia.org/wiki/Limite_(math%C3%A9matiques)#Limite_et_op%C3%A9rations_alg%C3%A9briques

@maple2015 

Using t = 12^(1/4)*omega^(1/2) you can estimate your zeros by fsolve( cos(t)*(cos(t)-sin(t)), t=desiredGuess), which has an approximate symbolic presentation (but needs a cleanup like shown by Christian Wolinski, evalc shows your function is real valued).

In you case tGuess = 5/4*Pi ~ 3.9269... or omegaGuess = 25/192*Pi^2*12^(1/2) ~ 4.45...

A "proof" for the zero to be minimal is through RootFinding:-Analytic (needs a cleanup for the original task).