Question

Solve the following equation:
${81}^{{\sin }^{2}x}+{81}^{{\cos }^{2}x}=30$

Answer 1

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

$({81}^{{\sin }^{2}x}{)}^2-30({81}^{{\sin }^{2}x})+81=0⟹{81}^{{\sin }^{2}x}=3,27⟹3^{4{\sin }^{2}x}=3,3^3$

${\sin }^{2}x=\frac{1}{4},\frac{3}{4}⟹x=n\pi +(-1{)}^n\frac{\pi }{6},m\pi +(-1{)}^m\frac{\pi }{3},$