Compute $\int_0^{\pi} \frac{\cos(nx)\cos(x) - \cos(nt)\cos(t)}{\cos(x) -\cos(t)}dt$

Let $n\in\mathbb{N}$ and $x\in]0,\pi[$, I am asked to calculate the following : $$ I_n = \int_0^{\pi} \dfrac{\cos(nx)\cos(x) - \cos(nt)\cos(t)}{\cos(x) -\cos(t)}dt$$ From testing on small values of $n$, it seems that this integral is equal to $n\pi\cdot \cos^n(x)$ but I can't seem to prove it. I tried finding a recurrence formula but didn't succeed.

Here is my working for $n=0$, $n=1$ and $n=2$ : For $n=0$, $$ I_0=\int_0^{\pi}\dfrac{\cos(x) -\cos(t)}{\cos(x) - \cos(t)}dt = \pi$$ For $n=1$, $$ I_1 = \int_0^{\pi} \dfrac{\cos^2(x) -\cos^2(t)}{\cos(x)-\cos(t)}dt=\int_0^{\pi}\cos(x) + \sin(t)dt = \pi\cdot \cos(x)$$ For $n=2$ : $$ I_2 = \int_0^{\pi} \dfrac{2\cos^3(x) - 2\cos^3(t) -\cos(x) + \cos(t)}{\cos(x) - \cos(t)}dt$$ $$ I_2 = 2\int_0^{\pi}\cos^2(x) +\cos(x)\cos(t) + cos^2(t) dt - \pi$$ $$ I_2 = 2\pi\cos^2(x) + \int_0^{\pi}\cos(2t)+1dt - \pi$$ $$ I_2 = 2\pi\cos^2(x) $$

This is my first post here, please tell me if I did anything wrong. I tried searching this integral on this website without any success.

Here's a solution that only rests on the following simple trigonometric identity: $$\cos(a+b)+\cos(a-b)=2\cos(a)\cos(b)\tag{1}$$ We'll get back to it later, but for now, notice that $$\begin{split} I_n(x)&=\int_0^{\pi} \frac{\cos(nx)\cos(x) - \cos(nt)\cos(t)}{\cos(x) -\cos(t)}dt\\ &=\int_0^{\pi}\frac{[\cos(nx)-\cos(nt)]\cos(x) + \cos(nt)[\cos(x)-\cos(t)]}{\cos(x) -\cos(t)}dt\\ &=\cos(x)\int_0^{\pi}\frac{\cos(nx)-\cos(nt)}{\cos(x) -\cos(t)}dt+\int_0^\pi\cos(nt)dt \end{split}$$ In other words, $$I_n(x)=\cos(x)J_n(x)+\pi\delta_{n=0}\tag{2}$$ where we define $$J_n(x)=\int_0^\pi \frac{\cos(nx)-\cos(nt)}{\cos(x)-\cos(t)}dt$$ and the Kronecker symbol $\delta_{n=0}$, which is equal $0$, unless $n=0$, in which case it's equal to $1$.

Now, let's go back to (1). Plugging $a=nx$ and $b=x$ into that identity implies that $$\cos((n+1)x)+\cos((n-1)x)=2\cos x \cos(nx)$$ Subtracting the same equation with $t$ to this one yields $$ \begin{split} \cos((n+1)x)-\cos((n+1)t) \\ +\cos((n-1)x)-\cos((n-1)t)=\\ 2\cos x \cos(nx)-2\cos(t)\cos(nt) \end{split}$$ Dividing by $\cos(x)-\cos(t)$, and integrating over $[0,\pi]$ leads to $$J_{n+1}(x)+J_{n-1}(x)=2I_n(x)\tag{3}$$ Finally, combining [2] and [3] gets us, for $n\geq 0$, $$J_{n+2}(x)-2\cos(x)J_{n+1}(x)+J_{n}(x)=0$$

The solution to this second-order recurrence relation is $$J_n(x)=\alpha e^{inx}+\beta e^{-inx}$$ Since, $J_0=0$ and $J_1=\pi$, $$J_n(x)=\frac {\pi \sin(nx)}{\sin x}$$ and $$I_n(x)=\pi\cos(x)\frac{\sin(nx)}{\sin(x)} \mbox{ for } n\geq 1 \mbox{, and }I_0=\pi$$

A nice way to evaluate generalized integrals is to consider them as the coefficients of an infinite series. Therefore, the coefficient of the $n$th term is simply the integral under question. Before we begin though, there is one identity to note:

$$\sum\limits_{n\geq0}z^n\cos nx=\frac {1-z\cos x}{z^2-2z\cos x+1}$$ Proof: Rewrite $\cos nx$ as the real part of $e^{nix}$. Using the infinite geometric sequences, we get that$$\sum\limits_{n\geq0}\left(ze^{ix}\right)^n=\frac 1{1-ze^{ix}}$$Now, take the real part of both sides. Clearly, the left - hand side becomes $z^n\cos nx$. Meanwhile, the right - hand side becomes, through some clever rationalization$$\begin{align*}\operatorname{Re}\left[\frac 1{1-ze^{ix}}\right] & =\operatorname{Re}\left[\frac 1{1-z\cos x-zi\sin x}\right]\\ & =\operatorname{Re}\left[\frac {1-z\cos x+zi\sin x}{(1-z\cos x)^2+z^2\sin^2x}\right]\\ & =\frac {1-z\cos x}{z^2-2z\cos x+1}\end{align*}$$completing the proof.

With that in mind, we are ready to begin. Since the OP has stated in the comments that he is trying to evaluate the integral$$I_n=\int\limits_0^{\pi}\mathrm dt\,\frac {\cos nx-\cos nt}{\cos x-\cos t}$$I will show a way to evaluate it in this answer. To wit, denote the generating function of the integral as $G(z)$

$$G(z)=\sum\limits_{n\geq0}I_nz^n$$

And remember that the coefficient of $z^n$ simply gives $I_n$. Interchange the sum and the integral, and using the identity we've derived above, get

$$\begin{align*}G(z) & =\int\limits_0^{\pi}\frac {\mathrm dt}{\cos x-\cos t}\sum\limits_{n\geq0}z^n\biggr[\cos nx-\cos nt\biggr]\\ & =\int\limits_0^{\pi}\frac {\mathrm dt}{\cos x-\cos t}\left[\frac {1-z\cos x}{z^2-2z\cos x+1}-\frac {1-z\cos t}{z^2-2z\cos t+1}\right]\end{align*}$$

Combining the two fractions, and recalling that any terms in $z$ are constants, the function becomes

$$G(z)=\frac {z(1-z^2)}{z^2-2z\cos x+1}\int\limits_0^{\pi}\frac {\mathrm dt}{z^2-2z\cos t+1}$$

The remaining integral can be easily evaluated using a Weierstrass substitution. Substitute $w=\tan\left(\tfrac t2\right)$ so that

$$\begin{array}{|c|c|c|}\hline w=\tan\left(\dfrac t2\right) & \mathrm dt=\dfrac {2\,\mathrm dw}{1+w^2} & \cos t=\dfrac {1-w^2}{1+w^2}\\\hline\end{array}$$

The remaining rational function can be evaluated in an elementary fashion

$$\begin{align*}G(z) & =\frac {2z(1-z^2)}{z^2-2z\cos x+1}\int\limits_0^{\infty}\frac {\mathrm dw}{w^2(1+z)^2+(1-z)^2}\\ & =\frac {2z}{z^2-2z\cos x+1}\arctan\left(\frac {1+z}{1-z}w\right)\,\Biggr\rvert_0^{\infty}\\ & =\frac {\pi z}{z^2-2z\cos x+1}\end{align*}$$

From the second line, it's important to observe that the argument of the inverse tangent will remain positive if and only if $|z|<1$. When $|z|<1$, then the denominator is positive, as well as the numerator. Therefore, $\tfrac {1+z}{1-z}>0$. However, if $|z|>1$, then the argument is less than zero and there is an extra negative sign. For the purpose of this question, we'll consider when $|z|<1$.

Now all we have to do is find the coefficient of $z^n$. There is a nice and convenient way to do this by using

$$2\cos x=e^{ix}+e^{-ix}$$

Factoring the denominator by grouping gives

$$\begin{align*}\frac z{z^2-2z\cos x+1} & =\frac z{(1-ze^{ix})(1-ze^{-ix})}\\ & =z\sum\limits_{k\geq0}z^k e^{kix}\sum\limits_{l\geq0}z^l e^{-lix}\end{align*}$$

Now observe what happens when we expand the products together$$\begin{multline}(1+ze^{ix}+z^2e^{2ix}+\cdots)(1+ze^{-ix}+z^2e^{-ix}+\cdots)\\=1+z(e^{ix}+e^{-ix})+z^2(e^{2ix}+1+e^{-2ix})+\cdots\end{multline}$$

The sum within the parenthesis seems to start off at the index of the $n$th term and decrease by a factor of two! Using this, it's possible to rewrite the coefficients conveniently as

$$a_k=\sum\limits_{m=0}^ke^{(k-2m)ix}=\frac {\sin x(k+1)}{\sin x}$$

Hence$$\frac {\pi z}{z^2-2z\cos x+1}=\pi\sum\limits_{k\geq1}\frac {\sin xk}{\sin x}z^k$$

And setting $k$ as $n$ gives the term $z^n$. Therefore, our integral is simply$$\int\limits_0^{\pi}\mathrm dt\,\frac {\cos nx-\cos nt}{\cos x-\cos t}\color{blue}{=\frac {\pi\sin xn}{\sin x}}$$

Completing Frank's solution:

$$ [z^n]\frac{\pi z}{z^2-2z\cos x+1} = \frac{\pi}{2}[z^{n}]\left(\frac{1}{z-e^{ix}}+\frac{1}{z-e^{-ix}}\right) $$ equals, by geometric series, $$ \frac{\pi}{2}\left(-e^{-(n+1)ix}-e^{(n+1)ix}\right)=-\pi\cos((n+1)x). $$

An alternative solution to the problem:

For $n \in \mathbb{N}$ and $x \in (0,\pi)$ define $$J_n (x) \equiv \int \limits_0^\pi \frac{\cos(n x) - \cos(n t)}{\cos(x) - \cos(t)} \, \mathrm{d} t \, . $$ We can use the identities ($(2)$ follows from the geometric progression formula) \begin{align} \cos(\xi) - \cos(\tau) &= - 2 \sin \left(\frac{\xi + \tau}{2}\right) \sin \left(\frac{\xi - \tau}{2}\right) \, , \, \xi,\tau \in \mathbb{R} \, , \tag{1} \\ \frac{\sin(n y)}{\sin(y)} &= \mathrm{e}^{-\mathrm{i}(n-1)y} \sum \limits_{k=0}^{n-1} \mathrm{e}^{2\mathrm{i} k y} \, , \, n \in \mathbb{N} \, , \, y \in \mathbb{R} \, , \tag{2} \\ \int \limits_0^{2 \pi} \mathrm{e}^{\mathrm{i}(k-l) t} \, \mathrm{d} t &= 2 \pi \delta_{k,l} \, , \, k,l \in \mathbb{Z} \, , \tag{3} \end{align} to compute \begin{align} J_n (x) &= \frac{1}{2} \int \limits_0^{2\pi} \frac{\cos(n x) - \cos(n t)}{\cos(x) - \cos(t)} \, \mathrm{d} t \stackrel{(1)}{=} \frac{1}{2} \int \limits_0^{2\pi} \frac{\sin \left(n\frac{x+t}{2}\right)}{\sin \left(\frac{x+t}{2}\right)} \frac{\sin \left(n\frac{x-t}{2}\right)}{\sin \left(\frac{x-t}{2}\right)} \, \mathrm{d} t \\ &\stackrel{(2)}{=} \frac{1}{2} \mathrm{e}^{-\mathrm{i} (n-1) x} \sum \limits_{k,l=0}^{n-1} \mathrm{e}^{\mathrm{i} (k+l) x} \int \limits_0^{2 \pi} \mathrm{e}^{\mathrm{i}(k-l) t} \, \mathrm{d} t \stackrel{(3)}{=} \pi \mathrm{e}^{-\mathrm{i} (n-1) x} \sum \limits_{k=0}^{n-1} \mathrm{e}^{2 \mathrm{i} k x} \\ &\stackrel{(2)}{=} \pi \frac{\sin(nx)}{\sin(x)} \, . \end{align} This result directly leads to \begin{align} I_n(x) &\equiv \int \limits_0^\pi \frac{\cos(n x) \cos(x) - \cos(n t) \cos(t)}{\cos(x) - \cos(t)} \, \mathrm{d} t = \int \limits_0^\pi \left[\cos(x)\frac{\cos(n x) - \cos(n t)}{\cos(x) - \cos(t)} + \cos(n t)\right]\, \mathrm{d} t \\ &= \cos(x) J_n(x) + 0 = \pi \cos(x) \frac{\sin(nx)}{\sin(x)} \, . \end{align}