Since $a=\sin^{-1}[\sin(-\large\frac{\pi}{4})]$$+\cos^{-1}[\cos\large\frac{2\pi}{3}]$
$\qquad\quad=\large\frac{-\pi}{4}+\frac{2\pi}{3}$
$\qquad\quad=\large\frac{5\pi}{12}$
$b=\tan^{-1}\big(\tan\big(-\large\frac{\pi}{3}\big)\big)$$-\cot^{-1}\big(\cot\big(\large\frac{2\pi}{3}\big)\big)$
$\;\;\;=-\large\frac{\pi}{3}-\frac{2\pi}{3}$
$\;\;\;=-\pi$
$a+b=\large\frac{5\pi}{12}$$-\pi$
$\qquad\;=-\large\frac{7\pi}{12}$
$a-b=\large\frac{5\pi}{12}$$+\pi$
$\qquad\;=\large\frac{17\pi}{12}$
Hence (a) is the correct option.