Explanation

(a)
We have, $\alpha + \beta + \gamma = \pi$
Now, $\sin^{2}{\alpha + \sin^{2}{\beta + sin^{2}\gamma}}$
$= \sin^{2}{\alpha + \sin^{2}{(\beta - \gamma)\sin{(\beta + \gamma)}}}$
$= \sin^{2}{\alpha + {sin\ }{(\pi - \alpha)\sin{(\beta + \gamma)}}}\ (\because\alpha + \beta - \gamma = \pi)$
$= \sin^{2}{\alpha + {sin\ \alpha\ }{\sin{(\beta + \gamma)}}}$
$= \sin{\alpha\left\lbrack \sin{\alpha + \sin{(\beta + \gamma)}} \right\rbrack}$
$= \sin{\alpha\left\lbrack {sin(}{\pi - (\beta - \gamma)) + \sin{(\beta + \gamma)}} \right\rbrack}$
$= \sin{\alpha\left\lbrack {sin(}{\beta - \gamma) + \sin{(\beta + \gamma)}} \right\rbrack}$
$= \sin{\alpha\ \left\lbrack 2\sin{\beta\cos\gamma} \right\rbrack}$
$= 2\sin{\alpha\sin{\beta\cos\gamma}}$