The number of distinct solutions of $\log_{1/2} |\sin x| = 2 - \log_{1/2} |\cos x|$ in $[0, 2\pi]$ is:
Step-by-Step Solution
Key Concept: Combine: $\log_{1/2}(|\sin x||\cos x|) = 2 \Rightarrow |\sin x \cos x| = \frac{1}{4} \Rightarrow |\sin 2x| = \frac{1}{2}$. In $[0, 2\pi]$: $2x \in [0, 4\pi]$, giving 8 solutions (4 for each sign of $\sin 2x = \pm \frac{1}{2}$). Verify none have $\sin x = 0$ or $\cos x = 0$.
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Correct Answer: (4)