Examine whether the folloiwng pair of function are identical or not: `f(x)=cosec^(2)(x)-cot^(2)x` and `g(x)=1`

Explanation

To determine whether the functions \( f(x) = \csc^2(x) - \cot^2(x) \) and \( g(x) = 1 \) are identical, we need to follow these steps:

### Step 1: Simplify \( f(x) \)

We start with the function \( f(x) \):
\[ f(x) = \csc^2(x) - \cot^2(x) \]

Using the Pythagorean identity for cosecant and cotangent, we know that:
\[ \csc^2(x) - \cot^2(x) = 1 \]
Thus, we can simplify \( f(x) \):
\[ f(x) = 1 \]

### Step 2: Compare the functions

Now we have:
\[ f(x) = 1 \quad \text{and} \quad g(x) = 1 \]
At first glance, both functions appear to be equal.

### Step 3: Check the domains of \( f(x) \) and \( g(x) \)

Next, we need to check the domains of both functions to see if they are identical.

1. **Domain of \( g(x) \)**:
- The function \( g(x) = 1 \) is defined for all real numbers \( x \).

2. **Domain of \( f(x) \)**:
- The function \( f(x) = \csc^2(x) - \cot^2(x) \) is defined wherever \( \csc(x) \) and \( \cot(x) \) are defined.
- The functions \( \csc(x) \) and \( \cot(x) \) are undefined at \( x = n\pi \) (where \( n \) is an integer), which means \( f(x) \) is also undefined at these points.

### Step 4: Conclusion

Since the domain of \( f(x) \) does not include all real numbers (it is undefined at \( x = n\pi \)), while the domain of \( g(x) \) includes all real numbers, we conclude that the two functions are not identical.

Thus, the final answer is:
\[ \text{The functions } f(x) \text{ and } g(x) \text{ are not identical.} \]

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