If $0\le x\le \pi /2$ and ${81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30$, then $x$is equal 

Solution:

Let ${81}^{{\sin }^{2}x}=y$ Than,

      ${81}^{{\cos }^{2}x}={81}^{1-{\sin }^{2}x}=81y^{-1}$

$\begin{array}{l}\Rightarrow {81}^{{\sin }^{2}x}+{51}^{{\cos }^{2}x}=30\\ \Rightarrow y+81y^{-1}=30\\ \Rightarrow y^2-30y+81=0\\ \Rightarrow y=3\text{ or, }y=27.\\ \Rightarrow {81}^{{\sin }^{2}x}=3\text{ or, }27\\ \Rightarrow 3^{4{\sin }^{2}x}=3^1\text{ or, }{3}^3\\ \Rightarrow 4{\sin }^{2}x=1\text{ or, }3\\ \Rightarrow {\sin }^{2}x=\frac{1}{4}\text{ or, }\frac{3}{4}\\ \Rightarrow \sin x=\pm \frac{1}{2}\text{ or, }\pm \frac{\sqrt{3}}{2}\Rightarrow x=\frac{\pi }{6}\text{ or, }\frac{\pi }{3}\end{array}$