It's All Evens and Odds

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

The area $A = 2n(\int_{0}^{\dfrac{\pi}{4n}} \tan^2(n\theta) d\theta + \int_{\dfrac{\pi}{4n}}^{\dfrac{\pi}{2n}} \cot^2(n\theta) d\theta) =$ $2n(\int_{0}^{\dfrac{\pi}{4n}} (\sec^2(n\theta) - 1) d\theta + \int_{\dfrac{\pi}{4n}}^{\dfrac{\pi}{2}}(\csc^2(n\theta) - 1) d\theta) =$ $2n(\dfrac{1}{n}\tan(n\theta) - \theta|_{0}^{\dfrac{\pi}{4n}} + -\dfrac{1}{n}\cot(n\theta) - \theta|_{\dfrac{\pi}{4n}}^{\dfrac{\pi}{2}}) = 2n(\dfrac{2}{n} - \dfrac{\pi}{2n}) = \boxed{4 - \pi} \approx \boxed{0.858407}$ .