Integrate the following :-
(3cos^2x+4sin^2x)/cos^2xsin^2x
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First, of course, 3cos^2(x)+ 4sin^2(x)= 3cos^2(x)+ 3sin^2(x)+ sin^2(x)= 3(cos^2(x)+ sin^2(x))+ sin^2(x)= 3+ sin^2(x). And the denominator is cos^2(x)sijn^2(x)= (1- sin^2(x))sin^2(x)= sin^2(x)- sin^4(x).
So the integrand can be written (3+ sin^2(x))/(sin^2(x)- sin^4(x)). Since those are all even powers of sin(x), I would use the identity sin^2(x)= (1- cos(2x))/2.