The number of roots of the equation, 81 sin 2 x + 81 cos 2 x = 30 in the interval 0 , π is equal to :

${\left(81\right)}^{{\sin }^{2}x}+{\left(81\right)}^{{\cos }^{2}x}=30$

${\left(81\right)}^{{\sin }^{2}x}+\frac{{\left(81\right)}^1}{{\left(81\right)}^{{\sin }^{2}x}}=30$

${\left(81\right)}^{{\sin }^{2}x}=t$

$t+\frac{81}{t}=30$

$t^2-30t+81=0$

$\left(t-3\right)\left(t-27\right)=0$

${\left(81\right)}^{{\sin }^{2}x}=3^1$ or  ${\left(81\right)}^{{\sin }^{2}x}=3^3$

$3^{4{\sin }^{2}x}=3^1$  or   $3^{4{\sin }^{2}x}=3^3$

${\sin }^{2}x=\frac{1}{4}$  or  ${\sin }^{2}x=\frac{3}{4}$

Hence, total $4$ solution.