how to evaluate $\int_0^\pi\frac{\cos(nx)-\cos(n\alpha)}{\cos(x)-\cos(\alpha)}dx$ without difference equations or special functions

Question

I saw this integral in this book inside interesting integrals

$$I_n=\int_0^\pi\frac{\cos(nx)-\cos(n\alpha)}{\cos(x)-\cos(\alpha)}dx$$ and the author of the book provided a very clever interesting way of evaluating this integral by showing that $I_{n+1}-2I_n\cos(\alpha)+I_{n-1}=0$

and then solving the difference equations

but as the author states

but I have no idea what motivated the person who first did this. The mystery of mathematical genius!

I was wondering if there is a another "forward ways" like Leibniz rule or some substitution to solve this problem and I couldn't get it to work with any other method.

$\cos(nx)$ and $\cos(n\alpha)$ are polynomials in $\cos(x)$ and $\cos(\alpha)$ respectively. In fact, they are the same polynomial, so $\cos(nx)-\cos(n\alpha)$ has a factor of $\cos(x)-\cos(\alpha)$. If you are willing to use Chebyshev polynomials, you can compute the general antiderivative. — Joshua Wang, Jul 03 '23 at 03:16
$$\int_{0}^{\pi}{\frac{\cos(nx)-\cos(na)}{\cos x-\cos a}}dx = \frac{\pi \sin(na )}{\sin a}$$ — Quanto, Jul 03 '23 at 03:28
It seems this one has been asked before. I can write a different method. — MathFail, Jul 03 '23 at 03:32
Does this answer your question? Parametric trigonometric integral $\int_{0}^{\pi}{\frac{\cos(nx)-\cos(na)}{\cos x-\cos a}}dx$ — MathFail, Jul 03 '23 at 03:39
@MathFail the 2 answers of the question one has a difference equation and the other use some type of polynomial that I never heard of , I asked if there is more "natural way" for solving the problem or a more motivated way — pie, Jul 03 '23 at 04:36
@MathFail as a some solution use methods that someone who just finished calc 3 book and an elementary real analysis book(me) can come up with and understand maybe some substitution , Leibniz rule of integration Euler formula for complex number , Taylor series ...etc — pie, Jul 03 '23 at 05:26
@MathFail I can see that the answer of my question is probably no because you and other great contributors on this side didn't answer the question instantly and moreover the question has been asked before and no one came with an easy solution — pie, Jul 03 '23 at 05:31

how to evaluate $\int_0^\pi\frac{\cos(nx)-\cos(n\alpha)}{\cos(x)-\cos(\alpha)}dx$ without difference equations or special functions

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