Tangents and Cotangents

Let $n$ be a positive integer and $n \geq 2$ .

The area $A = n(\displaystyle\int_{\frac{3\pi}{4n}}^{\frac{5\pi}{4n}} (1 + \tan(n\theta))^2 d\theta + \displaystyle\int_{\frac{5\pi}{4n}}^{\frac{7\pi}{4n}} (1 + \cot(n\theta))^2 d\theta) =$

$n(\displaystyle\int_{\frac{3\pi}{4n}}^{\frac{5\pi}{4n}} 2\tan(n\theta) + \sec^2(n\theta) d\theta + \displaystyle\int_{\frac{5\pi}{4n}}^{\frac{7\pi}{4n}} 2\cot(n\theta) + \csc^2(n\theta) d\theta) =$

$n((-\dfrac{2}{n}\ln|\cos(n\theta)| + \dfrac{1}{n}\tan(n\theta))|_{\frac{3\pi}{4n}}^{\frac{5\pi}{4n}} + (\dfrac{2}{n}\ln|\sin(n\theta) - \dfrac{1}{n}\cot(n\theta))|_{\frac{5\pi}{4n}}^{\frac{7\pi}{4n}}) =$ $n(\dfrac{4}{n}) = \boxed{4}$ .