Animated Solution for Mathematics - Trigonometry: The number of distinct solutions of the equation, ${\log }_{1/2}|\sin x|=2-{\log }_{1/2}|\cos x|$ in the interval $[0,2\pi ]$, is:

The equation provided is ${\log }_{1/2}|\sin x|=2-{\log }_{1/2}|\cos x|$. Before proceeding with algebraic manipulation, we must respect the domain of the logarithmic function.

The logarithm ${\log }_b(y)$ is defined only for $y>0$. Since we have $|\sin x|$ and $|\cos x|$, the absolute value ensures non-negativity, but the logarithm requires them to be strictly positive.

Therefore, we must satisfy $\sin x eq 0$ and $\cos x eq 0$. This excludes all quadrantal angles: $0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$. These are our forbidden zones that must be excluded to avoid 'ghost' solutions.

We begin by rearranging the equation to group the logarithmic terms:

Using the product rule for logarithms, ${\log }_{b}A+{\log }_{b}B={\log }_b(AB)$, we condense the expression:

Converting this into exponential form, we obtain:

To simplify further, we multiply both sides by $2$ to utilize the double-angle identity, $\sin 2x=2\sin x\cos x$:

This simplifies elegantly to:

This reduction transforms a complex logarithmic-trigonometric hybrid into a fundamental absolute value equation.

To determine the number of solutions in the interval $[0,2\pi ]$, we visualize the function $y=|\sin 2x|$. The period of $\sin 2x$ is $\pi$, meaning there are two full waves within the interval $[0,2\pi ]$.

The absolute value operation reflects the negative troughs of the sine wave upward, resulting in four positive humps within the interval $[0,2\pi ]$.

Drawing the horizontal line $y=\frac{1}{2}$ across these four humps reveals two intersection points per hump. Consequently, there are $4\times 2=8$ distinct solutions.

By trusting the geometry and respecting the domain, we have arrived at the final result of 8 solutions with absolute certainty.