To solve this type are throw graph. to $min ^m$, take lower portion.
=> $\int \limits_0^{\large\frac{\pi}{4}}$$ \tan x dx + \int \limits ^{\large\frac{\pi}{2}}_{\frac{\pi}{4}} \cot x dx$
=>$ \bigg[\log |\sec x|\bigg]_0^{\large\frac{\pi}{4}}+ \bigg[ \log |\sin x| \bigg]_{\frac{\pi}{4}}^{\frac{\pi}{2}}$
=> $\log | \sec \large\frac{\pi}{4}|$$ - | \sec 0| + \log | \sin \large\frac{z}{2}|$$ - \log \sin \large\frac{\pi}{2}$
=> $ \log \sqrt 2 -\log (1) + \log |1| - \log | \large\frac{1}{\sqrt 2}|$
=> $2 \log \sqrt 2$
=> $\log (\sqrt (2)^2)$
=> $ \log_e^2$