Question about the principal value of some integral

Question

So here is my problem,

is it possible that $$\int_{[0,1]}f(y)\cot(\pi(x-y))dy= p.v \int_{[0,1]}f(y)\cot(\pi(x-y))dy$$

I see that the left integral is singular for $x=0$ but since I never worked with the principal value I dont know if this equality is true.

The question appeared while I was trying to prove

Want to prove that the Hilbert transform of a $C^1(\mathbb T)$ function is the principal value of the convolution with $\cot(\pi x)$

Thanks