Which of the following pair of functions are identical ?

Options

A. `sqrt(1sinx),"sin"x/2+"cos"x/2`
B. `x,(x^(2))/x`
C. `sqrt(x^(2)),(sqrt(x))^(2)`
D. `lnx^(3)+lnx^(2),5lnx`

Explanation

To determine which pairs of functions are identical, we need to check if they have the same domain and range. Let's analyze each option step by step.

### Step 1: Analyze Option 1
**Functions:** \( f_1(x) = \sqrt{\sin x} \) and \( f_2(x) = \sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right) \)

- **Domain of \( f_1(x) = \sqrt{\sin x} \)**:
- For the square root to be defined, \( \sin x \) must be greater than or equal to 0.
- This occurs when \( x \) is in the intervals \( [2n\pi, (2n+1)\pi] \) for integers \( n \).

- **Domain of \( f_2(x) = \sin\left(\frac{x}{2}\right) + \cos\left(\frac{x}{2}\right) \)**:
- This function is defined for all real values of \( x \).

- **Conclusion for Option 1**: The domains are not the same, hence these functions are not identical.

### Step 2: Analyze Option 2
**Functions:** \( f_1(x) = x \) and \( f_2(x) = \frac{x^2}{x} \)

- **Domain of \( f_1(x) = x \)**:
- This function is defined for all real values of \( x \).

- **Domain of \( f_2(x) = \frac{x^2}{x} \)**:
- This function is defined for all real values of \( x \) except \( x = 0 \) (since division by zero is undefined).

- **Conclusion for Option 2**: The domains are not the same (as \( f_2 \) is undefined at \( x = 0 \)), hence these functions are not identical.

### Step 3: Analyze Option 3
**Functions:** \( f_1(x) = \sqrt{x^2} \) and \( f_2(x) = \sqrt{x^2} \)

- **Domain of \( f_1(x) = \sqrt{x^2} \)**:
- This function is defined for all real values of \( x \).

- **Domain of \( f_2(x) = \sqrt{x^2} \)**:
- This function is also defined for all real values of \( x \).

- **Conclusion for Option 3**: The domains are the same.

### Step 4: Analyze Option 4
**Functions:** \( f_1(x) = \ln(x^3) + \ln(x^2) \) and \( f_2(x) = \ln(x^5) \)

- **Domain of \( f_1(x) = \ln(x^3) + \ln(x^2) \)**:
- This function is defined for \( x > 0 \).

- **Domain of \( f_2(x) = \ln(x^5) \)**:
- This function is also defined for \( x > 0 \).

- **Range**: Both functions yield the same range as they are logarithmic functions.

- **Conclusion for Option 4**: The domains and ranges are the same, hence these functions are identical.

### Final Conclusion
The only pair of functions that are identical is from Option 4.