`f(x)=secx, g(x)=1/cosx` Identical or not?

Answer

To determine whether the functions \( f(x) = \sec x \) and \( g(x) = \frac{1}{\cos x} \) are identical, we need to follow these steps:

### Step 1: Define Identical Functions
Two functions \( f(x) \) and \( g(x) \) are said to be identical if they satisfy the following three conditions:
1. The domain of \( f(x) \) is equal to the domain of \( g(x) \).
2. The range of \( f(x) \) is equal to the range of \( g(x) \).
3. For every value of \( x \) in the domain of both functions, \( f(x) = g(x) \).

### Step 2: Determine the Domain of \( f(x) \)
The function \( f(x) = \sec x \) is defined as \( \sec x = \frac{1}{\cos x} \). The domain of \( \sec x \) excludes the values where \( \cos x = 0 \). This occurs at odd multiples of \( \frac{\pi}{2} \):
\[ \text{Domain of } f(x) = \{ x \in \mathbb{R} : x \neq (2n-1)\frac{\pi}{2}, n \in \mathbb{Z} \} \]

### Step 3: Determine the Domain of \( g(x) \)
The function \( g(x) = \frac{1}{\cos x} \) is also undefined where \( \cos x = 0 \). Thus, the domain of \( g(x) \) is the same as that of \( f(x) \):
\[ \text{Domain of } g(x) = \{ x \in \mathbb{R} : x \neq (2n-1)\frac{\pi}{2}, n \in \mathbb{Z} \} \]

### Step 4: Compare the Domains
Since both functions have the same domain:
\[ \text{Domain of } f(x) = \text{Domain of } g(x) \]
Thus, the first condition is satisfied.

### Step 5: Determine the Range of \( f(x) \)
The range of \( \sec x \) is \( (-\infty, -1] \cup [1, \infty) \) because \( \sec x \) takes values outside the interval \((-1, 1)\).

### Step 6: Determine the Range of \( g(x) \)
The range of \( g(x) = \frac{1}{\cos x} \) is also \( (-\infty, -1] \cup [1, \infty) \) for the same reason.

### Step 7: Compare the Ranges
Since both functions have the same range:
\[ \text{Range of } f(x) = \text{Range of } g(x) \]
Thus, the second condition is satisfied.

### Step 8: Check if \( f(x) = g(x) \)
For every \( x \) in the domain, we have:
\[ f(x) = \sec x = \frac{1}{\cos x} = g(x) \]
This holds true for all \( x \) in the domain of both functions.

### Conclusion
Since all three conditions are satisfied, we conclude that the functions \( f(x) \) and \( g(x) \) are identical:
\[ f(x) = g(x) \text{ for all } x \text{ in their common domain.} \]

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