$|\tan\theta+\sec\theta|=|\tan\theta|+|\sec\theta|, 0\leq \theta \leq 2\pi$ is possible only if-

$\mid \tan\theta+\sec\theta \mid=\mid \tan\theta \mid +\sec\theta$

Case 1:-

If $\theta \in [0, \cfrac{\pi}{2})$ then $\tan\theta=+ve , \,\, \sec\theta=+ve$

$\tan\theta+\sec\theta=+ve$

$\therefore \mid \sec\theta+\tan\theta \mid =\mid \tan\theta \mid +\mid \sec\theta \mid$

Case 2:-

$\theta \in (\cfrac{\pi}{2}, \pi]$

$\tan\theta=-ve , \,\, \sec\theta=-ve$

$\mid \sec\theta+\tan\theta \mid=+ve$

$\mid \sec\theta \mid=+ve \,\,\, \mid \tan\theta \mid=+ve$

$\mid \sec\theta+\tan\theta \mid=\mid \sec\theta \mid +\mid \tan\theta \mid$

Case 3:-

$\theta \in [ \pi,\cfrac{3\pi}{2}]$

$\tan\theta=+ve , \,\, \sec\theta=-ve$

$\mid \sec\theta+\tan\theta \mid=\tan\theta-\sec\theta$

$\mid \tan\theta \mid +\mid \sec\theta \mid=\sec\theta+\tan\theta$

$\mid \sec\theta+\tan\theta \mid \ne \mid \sec\theta \mid +\mid \tan\theta \mid$

Case 4:-

$\theta \in (\cfrac{3\pi}{2}, 2\pi]$

$\tan\theta=-ve , \,\, \sec\theta=+ve$

$\mid \sec\theta+\tan\theta \mid=\sec\theta -\tan\theta$

$\mid \sec\theta \mid=\sec\theta \,\,\, \mid \tan\theta \mid=\tan\theta$

$\mid \sec\theta+\tan\theta \mid \ne \mid \sec\theta \mid +\mid \tan\theta \mid$

$\theta \ne \cfrac{\pi}{2}$ because in that case $\tan\theta=\infty$