I am not sure I can give a general tip on proofs other than trying different ideas out. What do I mean by that?

Let's take this case. I see that what is to be proved is in terms of sine functions and what is given is in terms of sines and cosines, more specifically a cosine squared function. Do I know anything about the relationship about the squares of sines and cosines? I certainly do. So I also can "see" that I can eliminate the squared cosine and get everything in terms of sines and squares of sines. Does that help me? I don't know, but it certainly gives me something to try so

$\dfrac{sin^2(x) + 4sin(x) + 3}{cos^2(x)} = \dfrac{sin^2(x) + 4sin(x) + 3}{1 - sin^2(x)}.$

Here is a trick that is sometimes useful. I have a function that is a built up from a single function, in this case the sine function. It sometimes helps "seeing" how to proceed by simplifying using a substitution.

$Let\ u = sin(x).\ Then\ \dfrac{sin^2(x) + 4sin(x) + 3}{1 - sin^2(x)} = what.$

Can you simplify further now?