Integral $\int_0^\pi \cos (( n-m) x )\frac{\sin^2 (nmx)}{\sin(nx)\sin(mx)} \rm dx$
The value of the integral equals $\gcd (m,n)$ as stated in this article and the solution is not elemantary.
article citation: Sedjelmaci, Sidi Mohamed, Some related functions to integer GCD and coprimality, Bonomo, Flavia (ed.) et al., LAGOS’11 — VI Latin-American algorithms, graphs, and optimization symposium. Extended abstracts from the symposium, Bariloche, Argentina, March 28—April 1, 2011. Amsterdam: Elsevier. Electronic Notes in Discrete Mathematics 37, 135-140 (2011). ZBL1268.11162.
Maybe this is an elementary solution.
Via the formula \begin{equation*} a^p-b^p = (a-b)\sum_{k=0}^{p-1}a^{p-1-k}b^{k} \end{equation*} and Euler's formula we get \begin{gather*} \sin^2(nmx) = \left(\dfrac{e^{inmx}-e^{-inmx}}{2i}\right)^{2} =\dfrac{\left(e^{inx}\right)^{m}-\left(e^{-inx}\right)^{m}}{2i}\cdot \dfrac{\left(e^{imx}\right)^{n}-\left(e^{-imx}\right)^{n}}{2i}=\\[2ex] \dfrac{e^{inx}-e^{-inx}}{2i}\dfrac{e^{imx}-e^{-imx}}{2i}\cdot\sum_{k=0}^{m-1}e^{inx(m-1-k)}e^{-inxk}\cdot \sum_{j=0}^{n-1}e^{imx(n-1-j)}e^{-imxj}=\\[2ex] \sin(nx)\sin(mx)\cdot \sum_{k=0}^{m-1}\sum_{j=0}^{n-1}e^{ix(2nm-n-m-2kn-2jm)} \end{gather*} Thus \begin{gather*} \cos((n-m)x)\dfrac{\sin^2(nmx)}{\sin(nx)\sin(mx)}=\\[2ex]\dfrac{e^{i(n-m)x}+e^{-i(n-m)x}}{2}\cdot\sum_{k=0}^{m-1}\sum_{j=0}^{n-1}e^{ix(2nm-n-m-2kn-2jm)}=\\[2ex] \dfrac{1}{2}\cdot\sum_{k=0}^{m-1} \sum_{j=0}^{n-1}e^{ix(2nm-2m-2kn-2jm)}+ \dfrac{1}{2}\cdot\sum_{k=0}^{m-1} \sum_{j=0}^{n-1}e^{ix(2nm-2n-2kn-2jm)}.\tag{1} \end{gather*} However, \begin{equation*} \dfrac{1}{\pi}\int_{0}^{\pi}e^{i2px}\, \mathrm{d}x = \begin{cases} 0\text{ if } p\neq 0 \text{ and integer },\\ 1 \text{ if } p=0. \end{cases} \end{equation*} Now we integrate the two double sums in(1). By symmetry the two integrals will have the same value. The integral \begin{equation*} \dfrac{1}{\pi}\int_{0}^{\pi}e^{ix(2nm-2m-2kn-2jm)}\, \mathrm{d}x = 1 \end{equation*} if and only if \begin{gather*} nm-m-kn-jm=0 \tag{2} \end{gather*} which is a linear diophantine equaion.
Put $n= pd, m=qd$ where $d=\rm{gcd}(n,m)$. Then we can write (2) as \begin{equation*} kp+jq=pqd-q. \end{equation*} All solutions are \begin{equation*} \begin{cases} k=qd-rq\\ j=-1+rp \end{cases} \end{equation*} where $r$ is an integer. But $0 \le k \le m-1$ and $0\le j \le n-1$. Thus \begin{equation*} \begin{cases} 0\le qd-rq \le qd-1\\ 0 \le -1+rp \le pd -1 \end{cases} \Longleftrightarrow \begin{cases} 1\le rq \text{ and } r\le d\\ 1 \le rp \text{ and } r\le d. \end{cases} \end{equation*} Consequently we find $d$ solutions to (2). If we integrate (1) we get \begin{equation*} \dfrac{1}{2}d+\dfrac{1}{2}d = \rm{gcd(n,m).} \end{equation*}