Tranformation .

The required area is given by:

$\begin{aligned} A & = \int_0^\frac \pi 2 \big| |\tan x + \cot x | - |\tan x - \cot x| \big|\ dx \\ & = \int_0^\frac \pi 4 \big| |\tan x + \cot x | - |\tan x - \cot x| \big|\ dx + \int_\frac \pi 4^\frac \pi 2 \big| |\tan x + \cot x | - |\tan x - \cot x| \big|\ dx & \small \color{#3D99F6} \text{Note that } \tan x \le \cot x \text{ for }x \in \left[0, \frac \pi 4\right] \\ & = \int_0^\frac \pi 4 |\tan x + \cot x + \tan x - \cot x |\ dx + \int_\frac \pi 4^\frac \pi 2 | \tan x + \cot x - \tan x + \cot x |\ dx & \small \color{#3D99F6} \text{and } \tan x > \cot x \text{ for }x \in \left(\frac \pi 4, \frac \pi 2 \right] \\ & = 2 \int_0^\frac \pi 4 \tan x \ dx + 2 \int_\frac \pi 4^\frac \pi 2 \cot x \ dx \\ & = 2\ln(\cos x) \bigg|_\frac \pi 4^0 + 2 \ln (\sin x) \bigg|_\frac \pi 4^\frac \pi 2 \\ & = 2 \ln 2 \approx \boxed{1.386} \end{aligned}$