The number of roots of the equation, (81)^sin2x + (81)^cos2x = 30 in the interval [ 0, $\pi$ ] is equal to :

${(81)^{{{\sin }^2}x}} + {(81)^{1 - {{\sin }^2}x}} = 30$<br><br>${(81)^{{{\sin }^2}x}} + {{81} \over {{{(81)}^{{{\sin }^2}x}}}} = 30$<br><br>Let ${(81)^{{{\sin }^2}x}} = t$<br><br>$t + {{81} \over t} = 30 \Rightarrow {t^2} + 81 = 30t$<br><br>${t^2} - 30t + 81 = 0$<br><br>${t^2} - 27t - 3t + 81 = 0$<br><br>$(t - 3)(t - 27) = 0$<br><br>$t = 3,27$<br><br>${(81)^{{{\sin }^2}x}} = 3,{3^3}$<br><br>${3^{4{{\sin }^2}x}} = {3^1},{3^3}$<br><br>$4{\sin ^2}x = 1,3$<br><br>${\sin ^2}x = {1 \over 4},{3 \over 4}$<br><br>in [0, $\pi$ ] sin x &gt; 0<br><br>$\sin x = {1 \over 2},{{\sqrt 3 } \over 2}$<br><br>$x = {\pi \over 6},{{5\pi } \over 6},{\pi \over 3},{{2\pi } \over 3}$<br><br>Number of solution = 4