How do I prove $\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx = n\pi$? I need to show that the following integral:
$$\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx$$ =  $n\pi$, for all natural numbers $n$. What is the method to evaluate the above? 
I initially thought of the Leibniz rule,  but that wouldn't work as the result only holds for positive integers,  meaning the integral would not be a continuous function in $n$. Is there some other way to prove it?
 A: Here's another way.  Recall that $\sin(x) =\frac{e^{i x}-e^{-i x}}{2i}$.  Therefore
$$
I = \int_0^{\pi} \frac{\sin^2(nx)}{\sin^2(x)}\mathrm{d}x = \int_0^{\pi} \frac{\left(\frac{e^{inx}-e^{-inx}}{2i}\right)^2}{\left(\frac{e^{ix}-e^{-ix}}{2i}\right)^2} \mathrm{d}x =  \int_0^{\pi} \frac{e^{-i2nx}(e^{i2nx}-1)^2}{e^{-i2x}(e^{i2x}-1)^2} \mathrm{d}x = \int_0^{\pi} e^{-i2 (n-1)x}\frac{(e^{i2nx}-1)^2}{(e^{i2x}-1)^2} \mathrm{d}x
$$
For the next step, recall that $(t^n-1)/(t-1) = 1+t+\ldots+t^{n-1}$.  Using $t=e^{i2x}$ we get:
$$
I = \int_0^{\pi} e^{-i2(n-1)x}(1+e^{i2x}+\ldots+e^{i2(n-1)x})^2\mathrm{d}x 
$$
Then we multiply out to see that $(1+t+\ldots+t^{n-1})^2 = 1+2t+3t^2+\ldots nt^{n-1}+(n-1)t^{n+1}+\ldots+t^{2n-1}$ to see that the integral is
$$
I = \int_0^{\pi} e^{-i2(n-1)x}(1+\ldots+ne^{i2(n-1)x}+\ldots +e^{i2(2n-2)x})\mathrm{d}x
$$
Finally we note that if $n\neq 0$ then
$$
\int_0^\pi e^{i2nx}\mathrm{d}x = \left.\frac{1}{i2n}e^{i2nx}\right|_0^{\pi} = \frac{(-1)^{2n}-1}{i2n}=0 
$$
whereas for $n=0$ we get
$$
\int_0^\pi e^{i2nx}\mathrm{d}x  =
\int_0^\pi 1\mathrm{d}x = \pi
$$
Plugging into our expression from the integral we get
$$
I = n\pi.
$$
A: HINT.-A third quite different  way (no greater details for short).
$$I_n=\int_0^\pi\frac{(\sin nx)^2}{(\sin x)^2}dx=\int_0^\pi(\sin nx)^2)d(-\cot x)=n\pi$$
$$I_n=\left[(\sin nx)^2)(-\cot x)\right]_0^{\pi}+\int_0^\pi\frac{\sin 2nx(n\cos x)}{\sin x} =n\pi$$
The first term in RHS is equal to $0$ hence we can prove as equivalent statement that for all $n$ $$J_n=\int_0^\pi\frac{\sin 2nx\cos x}{\sin x} =\pi$$
It is verified that that $$J_1=\int_0^\pi\frac{\sin 2x\cos x}{\sin x} =\left[(x+\sin x\cos x\right]_0^{\pi}=\pi$$ so for finish we prove that $$J_{n+1}-J_n=0\qquad(*)$$ (the equality $(*)$ implies immediately that $J_2=J_3=\cdots J_n=\pi$)
We have the trigonometric identities
$$\begin{cases}\sin(2n+1)x\cos x=\frac 12\left[\sin(2n+2)x+\sin 2nx\right]\\\sin(2n)x\cos x=\frac 12[\sin(2n+1)x+\sin (2n-1)x]\end{cases}$$ It follows $$I_{n+1}-I_n=\frac 12\int_0^{\pi}\frac{(\sin(2n+2)x+\sin 2nx)-(\sin(2n+1)x+\sin (2n-1)x)}{\sin x}dx$$
We have here elementary integrals allow us to verify that $(*)$ is true.
A: If $I_n=\int_0^\pi\dfrac{\sin^2nx}{\sin^2x}dx$
$I_0=0,I_1=\pi$
$$I_{m+1}-I_m=\int_0^\pi\dfrac{\sin^2(m+1)x-\sin^2mx}{\sin^2x}dx=\int_0^\pi\dfrac{\sin(2m+1)x}{\sin x}dx=J_m\text{(say)}$$
Now $$J_{r+1}-J_r=\int_0^\pi\dfrac{\sin(2r+3)x-\sin(2r+1)x}{\sin x}dx=2\int_0^\pi\cos2(r+1)\ dx$$
For $r+1\ne0,$   $$J_{r+1}-J_r=\cdots=0$$
For $r\ge0,$
$$\implies J_r=J_0$$
$$J_0=I_1-I_0=\pi$$
For $m\ge0,$
$$I_{m+1}-I_m=J_0=\pi$$
Can you take it from here?
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
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 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
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\begin{align}
&\bbox[#ffd,10px]{\ds{\left.\int_{0}^{\pi}{\sin^{2}\pars{nx} \over \sin^{2}\pars{x}}\,\dd x\,\right\vert_{\ n\ \in\ \mathbb{Z}}}} =
{1 \over 2}\int_{-\pi}^{\pi}{\sin^{2}\pars{\verts{n}x} \over \sin^{2}\pars{x}}\,\dd x
\\[5mm] & =
{1 \over 2}\oint_{\verts{z}\ =\ 1^{-}}{\bracks{%
\pars{z^{\verts{n}} - z^{-\verts{n}}}/\pars{2\ic}}^{2} \over
\bracks{\pars{z - z^{-1}}/\pars{2\ic}}^{\,2}}\,{\dd z \over \ic z} =
{1 \over 2\ic}\oint_{\verts{z}\ =\ 1^{-}}{1 \over z^{2\verts{n} - 1}}
{\pars{1 - z^{2\verts{n}}}^{2} \over\pars{1 - z^{2}}^{\,2}}\,\dd z
\\[5mm] = &\
{1 \over 2\ic}\oint_{\verts{z}\ =\ 1^{-}}{1 \over z^{2\verts{n} - 1}}
\pars{1 - 2z^{2\verts{n}} + z^{4\verts{n}}}
\sum_{k = 0}^{\infty}\
\overbrace{{-2 \choose k}}^{\ds{\pars{k + 1}\pars{-1}^{k}}}\
\pars{-z^{2}}^{k}\,\dd z
\\[5mm] & =
\sum_{k = 0}^{\infty}\pars{k + 1}\bracks{%
\underbrace{{1 \over 2\ic}\oint_{\verts{z}\ =\ 1^{-}}{\dd z \over
z^{2\verts{n} - 2k - 1}}}_{\ds{\pi\,\delta_{2\verts{n} - 2k - 1,1}}} -
{1 \over 2\ic}\oint_{\verts{z}\ =\ 1^{-}}z^{2k + 1}\,\dd z +
{1 \over 2\ic}\oint_{\verts{z}\ =\ 1^{-}}z^{2\verts{n} + 2k + 1}\,\dd z}
\end{align}

The last two integrals vanish out.

Then,
$$
\bbx{\bbox[#ffd,10px]{\ds{\left.\int_{0}^{\pi}{\sin^{2}\pars{nx} \over \sin^{2}\pars{x}}\,\dd x\,\right\vert_{\ n\ \in\ \mathbb{Z}}}} = \verts{n}\pi}
$$
