2024/038) Solve for x $81^{\sin ^2x} + 81^{\cos^2x} = 30$

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

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

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

Let  $81^{\sin ^2x}=y$

So we get $y + \frac{81}{y}=30$

or $y^2-30y + 81=0$

or $(y-3)(y-27)=0$

Or $y=3$ or $y=27$

$y=3$ mean  $81^{\sin ^2x}=3$ or $sin^2x = \frac{1}{4}$ or $sin^2x = \pm \frac{1}{2}$ or $x= \pi \pm \frac{n\pi}{6}$

$y=27$ mean  $81^{\sin ^2x}=27$ or $sin^2x = \frac{3}{4}$ or $sin^2x = \pm \frac{\sqrt{3}}{2}$ or $x=  \pi \pm \frac{n\pi}{3}$

hence combining both we get $x= \frac{n\pi}{2}\pm \frac{\pi}{6}$