Question

If $xϵ[0,\pi ]$ and ${81}^{si{n}^{2}x}+{81}^{co{s}^{2}x}=30$, then find number of values of $x$ satisfy the given equation.

Answer 1

Let ${81}^{{\sin }^{2}x}=y$
${81}^{{\cos }^{2}x}={81}^{1-{\sin }^{2}x}=81y^{-1}$
So that the given equation can be written as
$y^2-30y+81=0\Rightarrow y=3$ or $y=27$
$\Rightarrow {81}^{{\sin }^{2}x}=3$ or 27
$\Rightarrow 3^{4{\sin }^{2}x}=3^1$ or $3^3$
$\Rightarrow 4{\sin }^{2}x=1$ or $3$
$\Rightarrow {\sin }^{2}x=\frac{1}{4}$ or $\frac{3}{4}$
$\Rightarrow \sin x=\pm \frac{1}{2}$ or $\pm \frac{\sqrt{3}}{2}$
$\Rightarrow x=\frac{\pi }{6},\frac{\pi }{3}$