Which of the following is an even function?

Options

A. `f(x) = cos[log(x+sqrt(x^(2)+1))],x in R`
B. `f(x) =(a^(x)-a^(-x))/(a^(x) + a^(-x))`, where `a gt 0, a ne 1, x in R`
C. `f(x) = log_(5)((1+x)/(1-x)), -1 lt x lt 1`
D. `f(x) = x^(3) + sin x, x in R`

Answer

To determine which of the given functions is an even function, we need to check each function to see if it satisfies the condition for even functions: \( f(-x) = f(x) \) for all \( x \).

### Step-by-Step Solution:

1. **Identify the Functions**:
Let's denote the functions as follows:
- \( f_1(x) = \cos(\log(x) + \sqrt{x^2 + 1}) \)
- \( f_2(x) = \frac{1^x - 1^{-x}}{1^x + 1^{-x}} \)
- \( f_3(x) = \log_5\left(\frac{1+x}{1-x}\right) \)
- \( f_4(x) = x^3 + \sin(x) \)

2. **Check \( f_1(x) \)**:
- Calculate \( f_1(-x) \):
\[ f_1(-x) = \cos(\log(-x) + \sqrt{(-x)^2 + 1}) = \cos(\log(-x) + \sqrt{x^2 + 1}) \]
- Since \( \log(-x) \) is not defined for positive \( x \), we cannot conclude anything about \( f_1(x) \) being even or odd.

3. **Check \( f_2(x) \)**:
- Calculate \( f_2(-x) \):
\[ f_2(-x) = \frac{1^{-x} - 1^{x}}{1^{-x} + 1^{x}} = \frac{- (1^x - 1^{-x})}{- (1^x + 1^{-x})} = -f_2(x) \]
- Since \( f_2(-x) = -f_2(x) \), \( f_2(x) \) is an odd function.

4. **Check \( f_3(x) \)**:
- Calculate \( f_3(-x) \):
\[ f_3(-x) = \log_5\left(\frac{1-x}{1+x}\right) \]
- This can be rewritten using properties of logarithms:
\[ f_3(-x) = -\log_5\left(\frac{1+x}{1-x}\right) = -f_3(x) \]
- Thus, \( f_3(x) \) is also an odd function.

5. **Check \( f_4(x) \)**:
- Calculate \( f_4(-x) \):
\[ f_4(-x) = (-x)^3 + \sin(-x) = -x^3 - \sin(x) \]
- This can be rewritten as:
\[ f_4(-x) = - (x^3 + \sin(x)) = -f_4(x) \]
- Therefore, \( f_4(x) \) is an odd function.

6. **Conclusion**:
- After checking all functions, we find that \( f_1(x) \) is not defined for negative inputs, and \( f_2(x) \), \( f_3(x) \), and \( f_4(x) \) are all odd functions.
- Thus, the only function that can be classified as even is \( f_1(x) \) if we consider its definition for positive \( x \).

### Final Answer:
The even function among the given options is \( f_1(x) = \cos(\log(x) + \sqrt{x^2 + 1}) \).