Question

If $0\le x\le\pi$ and $81^{\sin^{2}x}+81^{\cos^{2}x}=30$ Then $x$ takes the value}

• A. $\pi/6$
• B. $\pi/4$
• C. $3\pi/6$
• D. $2\pi/3$

Hint:

Substitute $a^{\sin^2 x}$ to convert the trigonometric equation into a quadratic one.

Answer 1

The Correct Option is A

Step 1: Concept
Let $81^{\sin^2 x} = t$. Since $\sin^2 x + \cos^2 x = 1$, $81^{\cos^2 x} = 81^{1 - \sin^2 x} = 81/t$.

Step 2: Analysis
$t + 81/t = 30 \implies t^2 - 30t + 81 = 0 \implies (t-27)(t-3) = 0$.

Step 3: Calculation
Case 1: $81^{\sin^2 x} = 3 \implies 3^{4 \sin^2 x} = 3^1 \implies 4 \sin^2 x = 1 \implies \sin^2 x = 1/4 \implies \sin x = 1/2$.
$x = \pi/6$.
Case 2: $81^{\sin^2 x} = 27 \implies 3^{4 \sin^2 x} = 3^3 \implies 4 \sin^2 x = 3 \implies \sin^2 x = 3/4 \implies \sin x = \sqrt{3}/2$.
$x = \pi/3$.

Step 4: Conclusion
From options, $\pi/6$ is correct. Final Answer: (A)