Challenging +1.8 This question requires multiple sophisticated steps: recognizing the trigonometric identity 1 + tan²θ = sec²θ, converting to logarithm properties, handling the domain restriction from -4sin2x > 0, converting sec²2x = 16sin²2x to cos²2x = sin²2x/16, and solving the resulting trigonometric equation. The combination of logarithms, trigonometric identities, domain analysis, and algebraic manipulation makes this significantly harder than average, though it follows a logical path once the key identity is spotted.
In this question you must show detailed reasoning.
Solve the following equation for \(x\) in the interval \(0° < x < 180°\)
$$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]
\textbf{In this question you must show detailed reasoning.}
Solve the following equation for $x$ in the interval $0° < x < 180°$
$$1 + \log_3\left(1 + \tan^2 2x\right) = 2\log_3(-4\sin 2x)$$ [8]
\hfill \mbox{\textit{SPS SPS FM 2025 Q13 [8]}}