On proving that the sum $\sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational using trigonometry.

Question

Prove that the sum $$ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$$ is irrational.

Well the problem though have good algaebric solution I was wondering whether it's possible to solve it using this approach or not ?

Let $(1000+k) = \tan(\theta)$ . Then,

$\sqrt{(1000+k)^2+1} = \sec(\theta)$

Similarly the required sum would be summation of n sec thetas for n distinct thetas .

But i can't proceed this way so help if possible.