Question

The number of solutions of the equation |cot x| = cot x + 1/sin x in the interval [0, 2π] is __________.

Hint:

Always check the domain constraints ($\sin x \neq 0$) and the case conditions when solving absolute value equations.

Answer 1

Correct Answer: 1

Step 1: Case 1: $\cot x \ge 0$. $\cot x = \cot x + \frac{1}{\sin x} \implies \frac{1}{\sin x} = 0$. No solution.
Step 2: Case 2: $\cot x<0$. $-\cot x = \cot x + \frac{1}{\sin x} \implies -2 \frac{\cos x}{\sin x} = \frac{1}{\sin x}$.
Step 3: Assuming $\sin x \neq 0$: $-2 \cos x = 1 \implies \cos x = -1/2$.
Step 4: In $[0, 2\pi]$, $\cos x = -1/2$ at $x = 2\pi/3$ (Quadrant II) and $x = 4\pi/3$ (Quadrant III).
Step 5: Check condition $\cot x<0$: At $x = 2\pi/3$, $\cot x$ is negative. (Valid) At $x = 4\pi/3$, $\cot x$ is positive. (Invalid)
Step 6: Only 1 solution exists ($x = 2\pi/3$). ]