If $x = a$ satisfy the equation $3^{\sin 2x + 2\cos^2 x} + 3^{1 - \sin 2x + 2\sin^2 x} = 28$, then $\sin(2a - \cos 2a)^2 + 8\sin 4a$ is equal to:
Step-by-Step Solution
Key Concept: Exponential equations reduce to quadratic form through substitution when symmetric terms appear.
Substituting $t = 3^{\sin 2x + 2\cos^2 x}$ transforms the equation $3^{\sin 2x + 2\cos^2 x} + \frac{3^3}{3^{\sin 2x + 2\cos^2 x}} = 28$ to $t + \frac{27}{t} = 28$, giving $t^2 - 28t + 27 = 0$ with solutions $t = 1, 27$. For $t=1$, $\sin 2x + 2\cos^2 x = 0$ yields $x = \frac{\pi}{2}, \frac{3\pi}{4}, \frac{7\pi}{4}$; for $t=27$, no solution exists since $\sin 2x + 2\cos^2 x \leq 3$.
Correct Answer: 1