Improve the numerator in the approximation by including a third-order term in it.

Question

Improve the numerator in the approximation by including a third-order term in it.

$\tan x\approx\frac{x}{\left(1-\frac{4x^{2}}{\pi^{2}}\right)\left(1+ax^{2}\right)}$

I can include a third-order term, but I don't know how to specify "a"

$ \tan x\approx\left(\frac{\pi}{2}-x\right)^{-1\ }$ $x\to\frac{\pi}{2}$

I will be grateful for the idea

Answer 1

Hint

Since $\tan(x)$ is odd you look for $b$ such that $$\tan x\sim\frac{x+b x^3}{\left(1-\frac{4x^{2}}{\pi^{2}}\right)\left(1+ax^{2}\right)}$$ Cross multiply, use Taylor series and cancel the coefficient of the first term where $b$ will appear. This will give $b$ as a function of $a$.