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 $a+b$.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → $\frac{2\pi}{3}$
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{3\pi}{4}$
$= \frac{12\pi}{12}$ =π
$b = \tan^{-1}(\sqrt{3}) - \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right)$
$ = \frac{\pi}{3} - \frac{2\pi}{3} = \frac{-\pi}{3}$
$\text{Now, } a + b $= π - $\frac{\pi}{3}$
$= \frac{2\pi}{3}$