$\text{ For }0\le \mathrm{x}\le \mathrm{\pi }\text{, if }{81}^{{\sin }^2\mathrm{x}}+{81}^{{\cos }^2\mathrm{x}}=30\text{, then }\mathrm{x}=$

${81}^{{\sin }^2\mathrm{x}}+{81}^{{\cos }^2\mathrm{x}}=30$

$\text{Put }{\cos }^{2}x=1-{\sin }^{2}x$

$\text{If}y={81}^{{\sin }^{2}x} \text{then }y+\frac{81}{y}=30$

$y^2-30y+81=0,y=27,3$

$3^{4{\sin }^{2}x}=3^3,3^1$

$\therefore {\sin }^{2}x=\frac{3}{4},\frac{1}{4}$

$\therefore \sin x=\frac{\sqrt{3}}{2},\frac{1}{2}$

$\text{as}\sin \mathrm{x}\text{is + ive in the given interval}0<\mathrm{x}<\pi$

$\therefore x=\frac{\pi }{3},\pi -\frac{\pi }{3}=\frac{2\pi }{3},\frac{\pi }{6},\pi -\frac{\pi }{6}=\frac{5\pi }{6}.$