Sometimes functions are defined like `f(x)=max{sinx,cosx}`, then `f(x)` is splitted like `f(x)={{:(cosx, x in (0,(pi)/(4)]),(sinx, x in ((pi)/(4),(pi)/(2)]):}` etc.
If `f(x)=min{tanx, cotx}` then `f(x)=1` when `x=`

Options

A. `(npi)/(2)+(pi)/(4)`
B. `npi+(pi)/(6)`
C. `npi+(pi)/(4)`
D. `2npi+(pi)/(4)`

Answer

To solve the problem where \( f(x) = \min\{\tan x, \cot x\} \) and we need to find when \( f(x) = 1 \), we can follow these steps:

### Step 1: Set the equation for \( f(x) \)
We have:
\[ f(x) = \min\{\tan x, \cot x\} \]
We want to find when \( f(x) = 1 \). This means we need to solve the equations:
\[ \tan x = 1 \quad \text{and} \quad \cot x = 1 \]

### Step 2: Solve \( \tan x = 1 \)
The equation \( \tan x = 1 \) holds true at:
\[ x = n\pi + \frac{\pi}{4} \]
where \( n \) is any integer.

### Step 3: Solve \( \cot x = 1 \)
The equation \( \cot x = 1 \) is equivalent to:
\[ \tan x = 1 \]
Thus, the solutions for \( \cot x = 1 \) are also:
\[ x = n\pi + \frac{\pi}{4} \]

### Step 4: Determine the intervals
Both equations give us the same general solution. Therefore, the function \( f(x) = 1 \) when:
\[ x = n\pi + \frac{\pi}{4} \]
for any integer \( n \).

### Conclusion
The values of \( x \) for which \( f(x) = 1 \) are given by:
\[ x = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \]