Test the function `f(x)=tan x +cot" x, where "0 lt x lt pi/2`, for increase and decrease.

Answer

To determine where the function \( f(x) = \tan x + \cot x \) is increasing or decreasing on the interval \( (0, \frac{\pi}{2}) \), we need to follow these steps:

### Step 1: Find the derivative of the function
To analyze the behavior of the function, we first find its derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}(\tan x) + \frac{d}{dx}(\cot x) \]

Using the derivatives of the trigonometric functions, we have:

\[ f'(x) = \sec^2 x - \csc^2 x \]

### Step 2: Rewrite the derivative in terms of sine and cosine
We can express \( \sec^2 x \) and \( \csc^2 x \) in terms of sine and cosine:

\[ f'(x) = \frac{1}{\cos^2 x} - \frac{1}{\sin^2 x} \]

### Step 3: Combine the fractions
To combine the fractions, we find a common denominator:

\[ f'(x) = \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} \]

### Step 4: Analyze the numerator
The sign of \( f'(x) \) depends on the numerator \( \sin^2 x - \cos^2 x \):

- \( f'(x) > 0 \) when \( \sin^2 x > \cos^2 x \)
- \( f'(x) < 0 \) when \( \sin^2 x < \cos^2 x \)

### Step 5: Determine critical points
The critical point occurs when \( \sin^2 x = \cos^2 x \), which happens at:

\[ \tan^2 x = 1 \quad \Rightarrow \quad x = \frac{\pi}{4} \]

### Step 6: Test intervals around the critical point
We need to test the sign of \( f'(x) \) in the intervals \( (0, \frac{\pi}{4}) \) and \( (\frac{\pi}{4}, \frac{\pi}{2}) \).

1. **For \( x \in (0, \frac{\pi}{4}) \)**:
- Choose \( x = \frac{\pi}{8} \):
- \( \sin^2(\frac{\pi}{8}) < \cos^2(\frac{\pi}{8}) \) (since \( \tan(\frac{\pi}{8}) < 1 \))
- Thus, \( f'(\frac{\pi}{8}) < 0 \) → \( f(x) \) is decreasing.

2. **For \( x \in (\frac{\pi}{4}, \frac{\pi}{2}) \)**:
- Choose \( x = \frac{3\pi}{8} \):
- \( \sin^2(\frac{3\pi}{8}) > \cos^2(\frac{3\pi}{8}) \) (since \( \tan(\frac{3\pi}{8}) > 1 \))
- Thus, \( f'(\frac{3\pi}{8}) > 0 \) → \( f(x) \) is increasing.

### Conclusion
- The function \( f(x) = \tan x + \cot x \) is decreasing on the interval \( (0, \frac{\pi}{4}) \).
- The function is increasing on the interval \( (\frac{\pi}{4}, \frac{\pi}{2}) \).