Explanation

(a)
As we know that
$|\sin^{- 1}{x| \leq \frac{\pi}{2}}$
$\therefore$ Given relation
$\sin^{- 1}{x + \sin^{- 1}{y + \sin^{- 1}{\mathcal{z =}\frac{3\pi}{2}}}}$
Is possible only when
$\sin^{- 1}{x + \sin^{- 1}{y = \sin^{- 1}{\mathcal{z =}\frac{\pi}{2}}}}$
$\Rightarrow x = y = \mathcal{z} = 1$
$\therefore x^{100} + y^{100} + \mathcal{z}^{100} - \frac{9}{x^{101} + y^{101} + \mathcal{z}^{101}}$
$= 1 + 1 + 1 - \frac{9}{1 + 1 + 1}$
$= 3 - \frac{9}{3} = 0$