Question

If $x\in [0,2\pi ]$, then the number of solutions of the equation ${81}^{{\cos }^{2}x}+{81}^{{\sin }^{2}x}=30$ is

Answer 1

${81}^{{\cos }^{2}x}+{81}^{{\sin }^{2}x}=30$
$\Rightarrow {81}^{{\sin }^{2}x}+{81}^{1-{\sin }^{2}x}=30$
$\Rightarrow {81}^{{\sin }^{2}x}+\frac{81}{{81}^{{\sin }^{2}x}}=30$
Let ${81}^{{\sin }^{2}x}=t$
Then $t+\frac{81}{t}=30$
$\Rightarrow t^2-30t+81=0$
$\Rightarrow (t-3)(t-27)=0$
$\Rightarrow t=3,27$

${81}^{{\sin }^{2}x}=3$ and ${81}^{{\sin }^{2}x}=27$
$\Rightarrow 3^{4{\sin }^{2}x}=3^1$ and $3^{4{\sin }^{2}x}=3^3$
$\Rightarrow {\sin }^{2}x=\frac{1}{4}$ and ${\sin }^{2}x=\frac{3}{4}$
$\Rightarrow \sin x=\pm \frac{1}{2}$ and $\sin x=\pm \frac{\sqrt{3}}{2}$

Since $x\in [0,2\pi ]$,
$\therefore x=\frac{\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{11\pi }{6},\frac{\pi }{3},\frac{2\pi }{3},\frac{4\pi }{3},\frac{5\pi }{3}$