Group Velocity of shallow water Stokes wave derivation seems wrong

[K29]

I have a simple question but I'm putting down the whole derivation as it is relevant. There is a point that I don't understand, or seems wrong to me. This is a derivation of Group Velocity followed by simplifying(approximating it) for long wavelength waves in shallow water. This appears in a pack of notes that I have. I feel like it is wrong. But if it is wrong, I don't see how to get to the right answer.

Group Velocity of Stokes wave in general is derived as such:

(1): V_{G}=\frac{d\omega(k)}{dk} where k is wavenumber, \omega is angular frequency

(2): \omega ^{2}=gk \tanh(kh_{0}). This is the dispersion relation for stokes waves. g is gravity, h_{0} is depth.

Differentiating both sides of 2 with respect to k:
2\omega \frac{d\omega}{dk}=g \tanh(kh_{0})+\frac{gkh_{0}}{\cosh ^{2}(kh_{0})}

Dividing through by 2\omega:
\frac{d\omega}{dk}=\frac{g}{2\omega}\tanh(kh_{0})+\frac{gkh_{0}}{\cosh ^{2}(kh_{0})}

Substituting in (2) rearranged:
\frac{d\omega}{dk}=\frac{g}{2\omega}\frac{\omega ^2}{gk}+\frac{gkh_{0}}{\cosh ^{2}(kh_{0})}=\frac{\omega}{2k}+\frac{gkh_{0}}{2\omega \cosh ^{2}(kh_{0})}

=\frac{\omega}{2k}[1+\frac{gh_{0}k^{2}}{\omega ^2 \cosh ^{2}(kh_{0})}]

Using phase velocity c=\frac{w}{k} substituting (2):
=\frac{c}{2}[1+\frac{gh_{0}k^{2}}{gk \tanh(kh_{0} \cosh ^{2}(kh_{0})}]

=\frac{c}{2}[1+\frac{gh_{0}k^{2}}{gk \sinh(kh_{0} \cosh(kh_{0})}]

Using 2\sinh(\theta) \cosh(\theta)=\sinh(2\theta)
(3):V_{G}=\frac{d\omega}{dk}=\frac{c}{2}[1+\frac{2kh_{0}}{\sinh(2kh_{0})}]

I am fine with all the steps up to there. I do not see any problems there.

Now we consider group velocity V_{G} with long wavelength, shallow water:
kh_{0}=2\pi\frac{h_{0}}{\lambda}<<1 since wave number k = 2pi/wavelength

We use the fact that
\sinh(x)=x+O(x^{3}) as x→0
(This step seems fine to me)

Using the above 2 facts in (3) we have:
4a.)V_{G}=\frac{c}{2}[1+\frac{2kh_{0}}{2kh_{0}}(1+O(kh_{0})^{2}]

This is where I have a problem. I see that a 2kh_{0} has been pulled out of O(kh_{0}^2). That is fine. But unless I am missing something the above equation only pops out if \sinh was in the numerator of (3). But it is in the denominator. So IN MY OPINION the above equation should be:
4.b)\frac{c}{2}[1+\frac{2kh_{0}}{2kh_{0}(1+O(kh_{0})^{2}}]

However as far as I can see 4.a) yields the correct result:
5)V_{G}=c[1+O(kh_{0})^2]\approx c

and 4b) does not.

Are the notes incorrect? If so, how does one arrive at 5.) If not, what am I missing?
PLEASE help.
Thanks

 

[clamtrox]

In (3), first expand sinh(x) and then 1/(1+x^2+...) using geometric series

 

[K29]

In (3), first expand sinh(x) and then 1/(1+x^2+...) using geometric series

Thanks a lot .That helped