Write down the domain of :
(i) `f(x) = sin^(-1) x `
(ii) ` f(x) = cos^(-1)x `
(iii) `f(x) = tan^(-1)x`
(iv) ` f(x) = sec^(-1) x `

Answer

To find the domain of the given inverse trigonometric functions, we will analyze each function step by step.

### (i) \( f(x) = \sin^{-1} x \)

1. The function \( \sin^{-1} x \) is defined for values of \( x \) where the sine function can take values.
2. The range of the sine function is from -1 to 1. Therefore, for \( f(x) \) to be defined, \( x \) must lie within this range.
3. Thus, the domain of \( f(x) = \sin^{-1} x \) is:
\[ \text{Domain} = [-1, 1] \]

### (ii) \( f(x) = \cos^{-1} x \)

1. The function \( \cos^{-1} x \) is defined for values of \( x \) where the cosine function can take values.
2. Similar to sine, the range of the cosine function is also from -1 to 1. Therefore, for \( f(x) \) to be defined, \( x \) must lie within this range.
3. Thus, the domain of \( f(x) = \cos^{-1} x \) is:
\[ \text{Domain} = [-1, 1] \]

### (iii) \( f(x) = \tan^{-1} x \)

1. The function \( \tan^{-1} x \) is defined for all real numbers \( x \).
2. There are no restrictions on the values of \( x \) for the tangent function.
3. Thus, the domain of \( f(x) = \tan^{-1} x \) is:
\[ \text{Domain} = (-\infty, \infty) \]

### (iv) \( f(x) = \sec^{-1} x \)

1. The function \( \sec^{-1} x \) is defined for values of \( x \) where the secant function can take values.
2. The range of the secant function is \( (-\infty, -1] \cup [1, \infty) \). Therefore, for \( f(x) \) to be defined, \( x \) must lie within this range.
3. Thus, the domain of \( f(x) = \sec^{-1} x \) is:
\[ \text{Domain} = (-\infty, -1] \cup [1, \infty) \]

### Summary of Domains:
- \( f(x) = \sin^{-1} x \) : Domain = \([-1, 1]\)
- \( f(x) = \cos^{-1} x \) : Domain = \([-1, 1]\)
- \( f(x) = \tan^{-1} x \) : Domain = \((- \infty, \infty)\)
- \( f(x) = \sec^{-1} x \) : Domain = \((- \infty, -1] \cup [1, \infty)\)