Which of the following pair of functions are identical

Options

A. `f(X)=sin^(2)x+cos^(2)x` and `g(x)=1`
B. `f(x)=sec^(2)x-tan^(2)x` and `g(x)=1`
C. `f(x)=cosec^(2)x-cot^(2)x` and `g(x)=1`
D. `f(x) =lnx^(2)` and `g(x)=2lnx`

Explanation

To determine which pair 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-by-Step Solution:

1. **Option 1:**
- Let \( f(x) = \sin^2 x + \cos^2 x \)
- Let \( g(x) = 1 \)
- **Domain of \( f(x) \)**: The function \( \sin^2 x + \cos^2 x \) is defined for all real numbers \( x \).
- **Domain of \( g(x) \)**: The constant function \( g(x) = 1 \) is also defined for all real numbers \( x \).
- **Range of \( f(x) \)**: The value of \( \sin^2 x + \cos^2 x \) is always 1 for all \( x \).
- **Range of \( g(x) \)**: The constant function \( g(x) = 1 \) has a range of {1}.
- **Conclusion for Option 1**: Both the domain and range are the same. Therefore, \( f(x) \) and \( g(x) \) are identical.

2. **Option 2:**
- Let \( f(x) = \frac{1}{x-2} \) (defined for all real \( x \) except \( x = 2 \))
- Let \( g(x) = x \) (defined for all real \( x \))
- **Domain of \( f(x) \)**: All real numbers except 2.
- **Domain of \( g(x) \)**: All real numbers.
- **Conclusion for Option 2**: The domains are not the same, so they are not identical.

3. **Option 3:**
- Let \( f(x) = \cos^2 x - \cot^2 x \)
- Let \( g(x) = \tan x \) (defined for all real \( x \) except \( x = n\pi + \frac{\pi}{2} \) for any integer \( n \))
- **Domain of \( f(x) \)**: The function \( \cot x \) is undefined where \( \sin x = 0 \), which occurs at \( x = n\pi \).
- **Domain of \( g(x) \)**: The function \( \tan x \) is defined for all real \( x \) except \( x = n\pi + \frac{\pi}{2} \).
- **Conclusion for Option 3**: The domains are not the same, so they are not identical.

4. **Option 4:**
- Let \( f(x) = \ln(x^2) \)
- Let \( g(x) = \ln(x) \) (defined for \( x > 0 \))
- **Domain of \( f(x) \)**: \( x^2 > 0 \) for all \( x \neq 0 \) (i.e., \( x \in \mathbb{R} \setminus \{0\} \)).
- **Domain of \( g(x) \)**: \( x > 0 \).
- **Conclusion for Option 4**: The domains are not the same, so they are not identical.

### Final Conclusion:
The only pair of functions that are identical is from **Option 1**: \( f(x) = \sin^2 x + \cos^2 x \) and \( g(x) = 1 \).