The solution of ${\left(81\right)}^{{\sin }^{2}x}+{\left(81\right)}^{{\cos }^{2}x}=30$ in  $[0,\frac{\pi }{2}]$ is

${81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30$, $[0,\frac{\pi }{2}]$

${81}^{{\sin }^{2}x}+{81}^{1-{\sin }^{2}x}=30$

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

${81}^{{\sin }^{2}x}=a$

$a+\frac{81}{a}=30$

$a^2+81-30a=0$

$a^2-30a+81=0$

$a^2-27a-3a+81=0$

$a(a-27)-3(a-27)=0$

$(a-27)(a-3)=0$

$a=27,a=3$

${81}^{{\sin }^{2}x}=3^3$

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

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

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

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

$\Rightarrow x=60$

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

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

$\Rightarrow x=30°$