Convergence test of the improper integral $\int_0^{\pi/2} \frac{1}{\cos\alpha-\cos x}dx$

Question

Consider the following problem:

Let $0<\alpha<\pi/2$,show that $\int_0^{\pi/2} \frac{1}{\cos\alpha-\cos x}dx$ diverges.

I am confused how to solve the above problem.I am not sure how to start with.Can someone provide me some clue for it?

I think it is discontinuous at $x=\alpha$.So we can split the integral at $\alpha$.

Answer 1

HINT:

Use the prosthaphaeresis identity

$$\cos(\alpha)-\cos(x)=2\sin\left(\frac{x+\alpha}{2}\right)\sin\left(\frac{x-\alpha}{2}\right)$$

and recall that $\sin(x)=O(x)$ as $x\to 0$ and $\frac1x$ is not integrable on a neighborhood of $0$.

Aside, the Cauchy Principal Value of the integral does exist for $\alpha \in (0,\pi/2)$.