If 0 ≤ x ≤ 2π, then the number of real values of x satisfying the...

Let $8{1}^{si{n}^{2}x} = t$,

then, $8{1}^{co{s}^{2}x} = 8{1}^{1 - si{n}^{2}x} = \dfrac{81}{t}$
So, the given equation is $t + \dfrac{81}{t} = 30$
${t}^{2} – 30t + 81 = 0$ $⇒ t = 3$ or $27$
$8{1}^{si{n}^{2}x} =$ $3$ or $27$
$⇒$ ${3}^{4si{n}^{2}x} = {3}^{1}$ or ${3}^{3}$
$⇒ \sin^{2} x = \dfrac{1}{4}$ or $\dfrac{3}{4} ⇒ sin x = \pm \dfrac{1}{2}, \pm \dfrac{\sqrt{3}}{2}$

Hence, there are $8$ solutions between $0$ and $2\pi$.