Explanation

(a)
Given, $81^{\sin^{2}x} + 81^{\cos^{2}x} = 30$
$\Rightarrow \ \ \ 81^{\sin^{2}x} + 81^{{1 - sin}^{2}x} = 30$
$\Rightarrow \ \ \ 81^{\sin^{2}x} + \frac{81}{81^{\sin^{2}x}} = 30$
$\Rightarrow \ \ y + \frac{81}{y} = 30\ \ \ \ \ \lbrack put\ 81^{\sin^{2}x} = y\rbrack$
$\Rightarrow \ \ y^{2} - 30y + 81 = 0$
$\Rightarrow \ \ (y - 27)(y - 3) = 0$
$\Rightarrow \ 81^{\sin^{2}x} = 27\$ or $81^{\sin^{2}x} = 3$
$\Rightarrow 3^{4\sin^{2}x\ \ } = 3^{3}$ or $3^{4\sin^{2}x} = 3$
$\Rightarrow \sin^{2}{x = \frac{3}{4}}$ or $\sin^{2}{x = \frac{1}{4}}$
$\Rightarrow \sin{x = \frac{\sqrt{3}}{2}}$ or $\sin{x = \frac{1}{2}}$
$\Rightarrow x = \frac{\pi}{3},\frac{2\pi}{3}\ \ or\ \ \ x = \frac{\pi}{6},\frac{5\pi}{6}$