Correct option is (2) 4
Given,
\((81)^{\sin^2x} + (81)^{\cos^2x} = 30\)
\((81)^{\sin^2x} + (81)^{1-\sin^2x} = 30\)
\(\Rightarrow (81)^{\sin^2x} + (81)^1 (81)^{-\sin^2x} = 30\)
\(\Rightarrow (81)^{\sin^2x} + \frac {81}{(81)^{\sin^2x}} = 30\)
Let \((81)^{\sin^2x}=t\)
\(\Rightarrow t + \frac{81}t = 30\)
\(⇒ t^2 - 30t + 81 = 0\)
\(⇒ (t - 27)(t - 3) = 0\)
\(⇒ t = 3 \ or\ t = 27\)
\(\Rightarrow (81)^{\sin^2x} = 3 \text{ to } (81)^{\sin^2x} = 27\)
\(\Rightarrow (3^4)^{\sin^2x} = 3 \text{ to } (3^4)^{\sin^2x} =3^3\)
\(\Rightarrow 3^{4\sin^2x} = 3^1 \text{ or } 3^{4\sin^2x} =3^3\)
\(⇒ 4\sin^2x = 1 \text{ or } 4\sin^2x = 3\)
\(\Rightarrow \sin^2x = \frac 14 \text{ or } \sin^2x = \frac 34\)
Now \(y = \sin x\)
For \(y = \sin^2x; x \in [0, \pi]\)
From the above figure, we can say that the given equation has 4 solution.
