Identify the following functions : `f(x)=xg(x)g(-x)+"tan"(sinx)`

Options

A. Odd
B. Even
C. Neither
D. Both

Explanation

To identify the function \( f(x) = x g(x) g(-x) + \tan(\sin x) \), we need to analyze its behavior under the transformation \( x \to -x \).

### Step-by-Step Solution:

1. **Write the function**:
\[ f(x) = x g(x) g(-x) + \tan(\sin x) \]

2. **Find \( f(-x) \)**:
Substitute \( -x \) into the function:
\[ f(-x) = -x g(-x) g(x) + \tan(\sin(-x)) \]

3. **Simplify \( \tan(\sin(-x)) \)**:
Using the property of sine, \( \sin(-x) = -\sin(x) \):
\[ \tan(\sin(-x)) = \tan(-\sin(x)) = -\tan(\sin(x)) \]

4. **Combine the terms**:
Now, substituting back into \( f(-x) \):
\[ f(-x) = -x g(-x) g(x) - \tan(\sin x) \]

5. **Factor out the negative sign**:
We can factor out a negative sign from the entire expression:
\[ f(-x) = -\left( x g(-x) g(x) + \tan(\sin x) \right) \]
This can be rewritten as:
\[ f(-x) = -f(x) \]

6. **Conclusion**:
Since \( f(-x) = -f(x) \), this indicates that the function \( f(x) \) is an **odd function**.

### Final Answer:
The function \( f(x) \) is an odd function.

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