Check whether pairs of function are identical or not ?
`f(x) = sec (sec^(-1) x) ` & g(x) `= cosec (cosec^(-1) x)`

Answer

To determine whether the functions \( f(x) = \sec(\sec^{-1}(x)) \) and \( g(x) = \csc(\csc^{-1}(x)) \) are identical, we will analyze each function step by step.

### Step 1: Analyze \( f(x) \)
The function \( f(x) = \sec(\sec^{-1}(x)) \).

- The secant inverse function, \( \sec^{-1}(x) \), is defined for \( x \geq 1 \) or \( x \leq -1 \).
- For \( x \) in this domain, \( \sec^{-1}(x) \) gives us an angle \( \theta \) such that \( \sec(\theta) = x \).
- Therefore, when we apply the secant function to this angle, we have:
\[ f(x) = \sec(\sec^{-1}(x)) = x \]
- Thus, \( f(x) \) is simply \( x \) for all \( x \) in the domain of \( \sec^{-1}(x) \).

### Step 2: Analyze \( g(x) \)
The function \( g(x) = \csc(\csc^{-1}(x)) \).

- The cosecant inverse function, \( \csc^{-1}(x) \), is defined for \( x \geq 1 \) or \( x \leq -1 \).
- For \( x \) in this domain, \( \csc^{-1}(x) \) gives us an angle \( \phi \) such that \( \csc(\phi) = x \).
- Therefore, when we apply the cosecant function to this angle, we have:
\[ g(x) = \csc(\csc^{-1}(x)) = x \]
- Thus, \( g(x) \) is also simply \( x \) for all \( x \) in the domain of \( \csc^{-1}(x) \).

### Step 3: Compare \( f(x) \) and \( g(x) \)
Now that we have both functions simplified:
- \( f(x) = x \) for \( x \geq 1 \) or \( x \leq -1 \)
- \( g(x) = x \) for \( x \geq 1 \) or \( x \leq -1 \)

Since both functions yield the same output for the same input in their respective domains, we conclude that:
\[ f(x) = g(x) \text{ for } x \geq 1 \text{ or } x \leq -1 \]

### Conclusion
The functions \( f(x) \) and \( g(x) \) are identical.

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