If `(secx)(tanx) lt 0`, which of the following must be true?
I. tan`xlt0`
II. Csc x cot `x lt 0`
III. X is in the third or fourth quadrant

Options

A. I only
B. II only
C. III only
D. II and III

Explanation

To solve the inequality \( \sec x \tan x < 0 \), we can follow these steps:

### Step 1: Rewrite the expression in terms of sine and cosine.
We know that:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]
Thus, we can rewrite the expression:
\[ \sec x \tan x = \frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} = \frac{\sin x}{\cos^2 x} \]

### Step 2: Set up the inequality.
We need to solve the inequality:
\[ \frac{\sin x}{\cos^2 x} < 0 \]

### Step 3: Analyze the signs of the numerator and denominator.
- The term \( \cos^2 x \) is always non-negative (since it is a square). Therefore, the sign of the entire expression \( \frac{\sin x}{\cos^2 x} \) depends solely on the sign of \( \sin x \).
- For the expression to be less than zero, \( \sin x \) must be negative:
\[ \sin x < 0 \]

### Step 4: Determine the quadrants where \( \sin x < 0 \).
The sine function is negative in the third and fourth quadrants. Therefore:
- \( x \) lies in the third quadrant (where \( \sin x < 0 \)) or the fourth quadrant (where \( \sin x < 0 \)).

### Step 5: Evaluate the options.
Now, let's analyze the given statements:
1. **I. \( \tan x < 0 \)**: This is true in the second and fourth quadrants, but not necessarily in the third quadrant. So, this statement is not always true.
2. **II. \( \csc x \cot x < 0 \)**: This is equivalent to \( \frac{1}{\sin x} \cdot \frac{\cos x}{\sin x} < 0 \) or \( \frac{\cos x}{\sin^2 x} < 0 \). Since \( \sin x < 0 \) and \( \cos x \) can be either positive or negative depending on the quadrant, this statement is not necessarily true.
3. **III. \( x \) is in the third or fourth quadrant**: This is true since we established that \( \sin x < 0 \) holds true in these quadrants.

### Conclusion:
The only statement that must be true is:
- **III. \( x \) is in the third or fourth quadrant.**

### Final Answer:
The correct answer is **III only**.

---