The number of distant solutions of the equation,$$log12sinx=2$$−$$log12cosx$$ in the interval , $$0$$,$$2$$$$π$$$$is______________$$

$$log1/2$$|sinx|$$=2-log1/2$$|cosx|$$log1/2$$|sinxcosx|$$=2$$|sinxcosx|$$=14sin2x=$$±$$12$$,$$2x$$∈[$$0$$,$$4$$π]

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Question

The number of distant solutions of the equation, $\log_{\frac{1}{2}} |\sin x| = 2 - \log_{\frac{1}{2}} |\cos x|$ in the interval , $[0, 2 π]$ is______________

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Solution

$\begin{array}{l} \log_{1 / 2} | \sin x | = 2 - \log_{1 / 2} | \cos x | \\ \log_{1 / 2} | \sin x \cos x | = 2 \\ | \sin x \cos x | = \frac{1}{4} \end{array}$
$\sin 2 x = \pm \frac{1}{2}, 2 x \in [0, 4 π]$

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