Step-by-step Solution:

Both the curves intersect at \(x = \frac{\pi}{4}\), and the enclosed region is denoted as **ACBDA**. Hence, the area of the enclosed region is: \[ \int_0^{\frac{\pi}{4}} \tan x \, dx + \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cot x \, dx \] This simplifies to: \[ = \left[ \log \sec x \right]_0^{\pi/4} + \left[ \log \sin x \right]_{\pi/4}^{\pi/2} \] Substitute the limits of integration: \[ = \left[ \log \sqrt{2} - \log 1 \right] + \left[ \log 1 - \log \frac{1}{\sqrt{2}} \right] \] Combine the terms: \[ = \log 2 \]