Question:
The number of roots of the equation,
$(81)^{\sin ^{2} x}+(81)^{\cos ^{2} x}=30$
in the interval $[0, \pi]$ is equal to:
Correct Option: , 2
Solution:
$(81)^{\sin ^{2} x}+(81)^{\cos ^{2} x}=30$
$(81)^{\sin ^{2} x}+\frac{(81)^{1}}{(18)^{\sin ^{2} x}}=30$
$(81)^{\sin ^{2} x}=t$
$\mathrm{t}+\frac{81}{t}=30$
$(t-3)(t-27)=0$
$(81)^{\sin ^{2} x}=3^{1} \quad$ or $\quad(81)^{\sin ^{2} x}=3^{3}$
$3^{4 \sin ^{2} x}=3^{1} \quad$ or $\quad 3^{4 \sin ^{2} x}=3^{3}$
$\sin ^{2} x=\frac{1}{4} \quad$ or $\quad \sin ^{2} x=\frac{3}{4}$
