Which one of the following does not reduce to sinx for every x, wherever defined, is

Options

A. `(tanx)/(secx)`
B. `(sinx)/(sec^(2)x-tan^(2)x)`
C. `(sin^(2)xx sec xx)/(tanx)`
D. All reduce to sinx

Answer

To determine which of the given functions does not reduce to sin(x) for every x where defined, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Identify the Functions**: We have four functions to analyze. Let's denote them as:
- Option A: \( \frac{\tan x}{\sec x} \)
- Option B: \( \frac{\sin x}{\sec^2 x - \tan^2 x} \)
- Option C: \( \sin^2 x \cdot \sec x \cdot \tan x \)
- Option D: (not specified in the transcript, but we will assume it is another function).

2. **Analyze Option A**:
\[ \frac{\tan x}{\sec x} = \frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}} = \sin x \]
- **Conclusion**: Option A reduces to sin(x).

3. **Analyze Option B**:
\[ \frac{\sin x}{\sec^2 x - \tan^2 x} \]
- We know from the identity:
\[ \sec^2 x - \tan^2 x = 1 \]
- Therefore,
\[ \frac{\sin x}{1} = \sin x \]
- **Conclusion**: Option B reduces to sin(x).

4. **Analyze Option C**:
\[ \sin^2 x \cdot \sec x \cdot \tan x = \sin^2 x \cdot \frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} = \frac{\sin^3 x}{\cos^2 x} \]
- This does not simplify to sin(x) for all x, as it depends on the values of sin(x) and cos(x).
- **Conclusion**: Option C does not reduce to sin(x).

5. **Analyze Option D**:
- Since the function for Option D is not specified in the transcript, we cannot analyze it. However, based on our findings, we can conclude that Option C is the only one that does not reduce to sin(x).

### Final Answer:
The function that does not reduce to sin(x) for every x, wherever defined, is **Option C**.