The total number of solutions of $\tan\{x\} = \cot\{x\}$ ; where $\{x\}$ denotes the fractional part of $x$ in $[0, 2\pi)$ is _______.
Step-by-Step Solution
Key Concept: The fractional part function $\{x\}$ has period 1, and finding intersections with a horizontal line $y = \frac{\pi}{4}$ requires counting crossings over the given interval.
From $\tan\{x\} = \cot\{x\}$ with $\{x\} \in (0,1)$, we get $\tan^2\{x\} = 1$, so $\tan\{x\} = 1$ (since tangent is positive on this interval). Thus $\{x\} = \tan^{-1}(1) = \frac{\pi}{4}$. For $x \in [0, 2\pi]$, the graphs of $y = \{x\}$ and $y = \frac{\pi}{4}$ intersect at 6 points as shown in the diagram.
Correct Answer: 6