Explanation

(c)
Now$,\sin{(\alpha - \beta) = \sin{(\theta - \beta - (\theta - \alpha)}}$)
$= \sin{(\theta - \beta) = \cos{(\theta - \alpha) - \cos{(\theta - \beta)\sin{(\theta - \alpha)}}}}$
$= ba - \sqrt{1 - b^{2}}\sqrt{1 - a^{2}}$
and$\ {cos\ }{(\alpha - \beta) = \cos{(\theta - \beta - (\theta - \alpha))}}$
$= \cos{(\theta - \beta)\cos{(\theta - \alpha) + \sin{(\theta - \beta)\sin{(\theta - \alpha)}}}}$
$= a\sqrt{1 - b^{2}} + b\sqrt{1 - a^{2}}$
$\therefore\cos^{2}{(\alpha - \beta) + 2ab\sin{(\alpha - \beta)}}$
$= (a\sqrt{1 - b^{2}} + b{\sqrt{1 - a^{2})}}^{2} + 2ab(ab - \sqrt{1 - a^{2}}\sqrt{1 - b^{2})}$
$= a^{2} + b^{2}$