Determine whether the following integral converges or diverges
$\int_{0}^{\pi/2}\frac{1}{sin^2(x)cos^2(x)}$
And assuming I have already proved the comparison tests, and right now I can use comparison test to prove the integral diverges
proof:$\int_{0}^{\pi/2}\frac{1}{sin^2(x)cos^2(x)}\ge \int_{0}^{\pi/2}\frac{sin^2(x)}{sin^2(x)cos^2(x)}=\int_{0}^{\pi/2}\frac{1}{cos^2(x)}=\int_{0}^{\pi/2}sec^2(x)=\left[tan(x)\right]_0^{\pi/2}=\infty$
for which clearly the original intergal diverges.
Is my proof correct? If it is wrong, where are the problems?
The problem of your proof is simple: from the statement itself, it is not guaranteed that $$\int_0^{\frac{\pi}{2}}\frac{1}{\sin^2(x)\cos^2(x)}dx$$ is a well-defined object. You have to show the integration is well-defined first.
Of course, if you did the homework correctly, you would surely show that we can break down the interval into two, and thus on each interval there's only one singular point, so that the improper integral is defined according to our textbook definition. (By the way, you did not submit this homework). During the test, you can only use the statement so you have to repeat this step to use the comparison test.
After this class, you can surely do problems in your way, which is mathematically correct; in this test, however, I have to stick to the rubric. Eventually, you would find this tedious, but from my memory, your reasonings in exams are very sloppy. I definitely think the formal reasoning is a vital part of the course, and I hope you can practice a bit more on this in the future.