Check whether following pairs of function are identical or not?
`f(x)=tanx` and `g(x)=1/(cotx)`

Explanation

To determine whether the functions \( f(x) = \tan x \) and \( g(x) = \frac{1}{\cot x} \) are identical, we need to check their domains and ranges.

### Step 1: Find the Domain of \( f(x) \)

The function \( f(x) = \tan x \) is defined as:
\[ f(x) = \frac{\sin x}{\cos x} \]
The tangent function is undefined when \( \cos x = 0 \). This occurs at:
\[ x = \frac{\pi}{2} + n\pi \quad \text{where } n \in \mathbb{Z} \]
Thus, the domain of \( f(x) \) is:
\[ \text{Domain of } f(x) = \mathbb{R} - \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} \]

### Step 2: Find the Domain of \( g(x) \)

The function \( g(x) = \frac{1}{\cot x} \) can be rewritten as:
\[ g(x) = \frac{1}{\frac{\cos x}{\sin x}} = \frac{\sin x}{\cos x} = \tan x \]
The cotangent function \( \cot x \) is undefined when \( \sin x = 0 \). This occurs at:
\[ x = n\pi \quad \text{where } n \in \mathbb{Z} \]
Thus, the domain of \( g(x) \) is:
\[ \text{Domain of } g(x) = \mathbb{R} - \{ n\pi \mid n \in \mathbb{Z} \} \]

### Step 3: Compare the Domains

The domains of \( f(x) \) and \( g(x) \) are:
- Domain of \( f(x) \): \( \mathbb{R} - \left\{ \frac{\pi}{2} + n\pi \mid n \in \mathbb{Z} \right\} \)
- Domain of \( g(x) \): \( \mathbb{R} - \{ n\pi \mid n \in \mathbb{Z} \} \)

Since the points where \( f(x) \) and \( g(x) \) are undefined are different, their domains are not the same.

### Step 4: Find the Range of \( f(x) \)

The range of \( f(x) = \tan x \) is:
\[ \text{Range of } f(x) = (-\infty, \infty) \]

### Step 5: Find the Range of \( g(x) \)

Since \( g(x) = \tan x \), its range is also:
\[ \text{Range of } g(x) = (-\infty, \infty) \]

### Step 6: Compare the Ranges

Both functions have the same range:
- Range of \( f(x) \): \( (-\infty, \infty) \)
- Range of \( g(x) \): \( (-\infty, \infty) \)

### Conclusion

Since the domains of \( f(x) \) and \( g(x) \) are not the same, we conclude that the functions \( f(x) \) and \( g(x) \) are **not identical**.