We are asked to solve the integral:
[tex]{:\implies \quad \displaystyle \sf \int \dfrac{dx}{\cos^{2}(x)-\tan (x)\cos^{2}(x)}}[/tex]
Re write as
[tex]{:\implies \quad \displaystyle \sf \int \dfrac{dx}{\cos^{2}(x)\{1-\tan (x)\}}}[/tex]
Using (1/cos x) = sec(x), we have
[tex]{:\implies \quad \displaystyle \sf \int \dfrac{\sec^{2}(x)dx}{1-\tan (x)}}[/tex]
Now, substitute 1 - tan (x) = t, so that -dt = sec²(x) dx
[tex]{:\implies \quad \displaystyle \sf -\int \dfrac{1}{t}dt}[/tex]
[tex]{:\implies \quad \sf log|t|+C}[/tex]
[tex]{:\implies \quad \boxed{\displaystyle \bf \int \dfrac{dx}{\cos^{2}(x)-\tan (x)\cos^{2}(x)}=-log|1-\tan (x)|+C}}[/tex]
Where, C is any Arbitrary Constant
