If $a = \sin^{-1} \left( \frac{\sqrt{2}}{2} \right) + \cos^{-1} \left( -\frac{1}{\sqrt{2}} \right)$ and $b = \tan^{-1}(\sqrt{3}) - \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right)$, then find the value of $a+b$.

$\frac{7\pi}{12}$

The correct answer is Option (1) → $\frac{7\pi}{12}$ ##

Since $a = \sin^{-1} \left( \frac{\sqrt{2}}{2} \right) + \cos^{-1} \left( -\frac{1}{\sqrt{2}} \right)$

$ = \sin^{-1} \left( \frac{1}{\sqrt{2}} \right) + \cos^{-1} \left( \frac{-1}{2} \right)$

$ = \frac{\pi}{4} + \frac{2\pi}{3}$

$= \frac{11\pi}{12}$

$b = \tan^{-1}(\sqrt{3}) - \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right)$

$ = \frac{\pi}{3} - \frac{2\pi}{3} = \frac{-\pi}{3}$

$\text{Now, } a + b = \frac{11\pi}{12} - \frac{\pi}{3}=\frac{11\pi-4\pi}{12}$

$= \frac{7\pi}{12}$