How to prove main-lobe side-lobe ratio in hann window is $-32\textrm{ dB}$?
So, the Hann function is defined to be [1, eq. 27b]

$$w(t)= \sin^2 \left ( \frac{ \pi t}{N-1} \right)\tag{def}$$

Now, to find local maxima, you'd just go ahead,


find the Fourier transform $W(f)$ of that (easy: it's a multiplication of two sines and a rectangular window in time domain, which means it's the convolution of the transforms of these sines and the window), and
do like every 11th grader and derive $\frac{dW(f)}{df}$ (harder, since now includes deriving a sum of sincs, but I'm pretty sure you'll manage!)
set that function, its absolute or its square  to 0 (hint: for which values does a cosine have a zero value?),
find the point $f_\text{sidelobe}$ of the first local maximum that's not at $f=0$ (which is the main lobe) , and insert that point in the original window, giving you a value for $W(f_\text{sidelobe})$.


Things can get arbitrarily complicated as soon as you realize that sidelobes that are at $t\notin \mathbb Z$ kind of don't count, but presuming this is an assignment and not a dissertation... assume a very very large $N$ in the worst case.

Alternative: Read [1], which very nicely derives both the spectrum of the Hann Window (by presenting it as a sum of three Dirichlet Kernels) and why it's very elegant in multiple aspects. 


[1] harris, Frederic J.:"On the use of windows for harmonic analysis with the discrete Fourier transform," in Proceedings of the IEEE, vol. 66, no. 1, pp. 51-83, Jan. 1978.
doi: 10.1109/PROC.1978.10837