The number of rational numbers in the domain of function
`y=sin^(-1)x+cos^(-1)x+tan^(-1)x+"cosec"^(-1)x+sec^(-1)x+cot^(-1)x`, is ___________ .

Answer

To find the number of rational numbers in the domain of the function

\[ y = \sin^{-1}x + \cos^{-1}x + \tan^{-1}x + \csc^{-1}x + \sec^{-1}x + \cot^{-1}x, \]

we need to determine the domain of each individual inverse trigonometric function and then find the intersection of these domains.

### Step 1: Determine the domain of each function

1. **Domain of \(\sin^{-1}x\)**:
- The domain is \(x \in [-1, 1]\).

2. **Domain of \(\cos^{-1}x\)**:
- The domain is \(x \in [-1, 1]\).

3. **Domain of \(\tan^{-1}x\)**:
- The domain is all real numbers, \(x \in (-\infty, \infty)\).

4. **Domain of \(\csc^{-1}x\)**:
- The domain is \(x \in (-\infty, -1] \cup [1, \infty)\).

5. **Domain of \(\sec^{-1}x\)**:
- The domain is \(x \in (-\infty, -1] \cup [1, \infty)\).

6. **Domain of \(\cot^{-1}x\)**:
- The domain is all real numbers, \(x \in (-\infty, \infty)\).

### Step 2: Find the intersection of the domains

Now, we need to find the intersection of all these domains:

- The common domain for \(\sin^{-1}x\) and \(\cos^{-1}x\) is \([-1, 1]\).
- The domains of \(\csc^{-1}x\) and \(\sec^{-1}x\) are \((- \infty, -1] \cup [1, \infty)\).

To find the intersection:

- The intersection of \([-1, 1]\) with \((- \infty, -1] \cup [1, \infty)\) gives us the points where these intervals overlap.
- The only points in \([-1, 1]\) that also lie in \((- \infty, -1] \cup [1, \infty)\) are \(-1\) and \(1\).

### Step 3: Identify the rational numbers in the domain

The rational numbers in the intersection set \([-1, 1]\) are:

- \(-1\) (which is rational)
- \(1\) (which is also rational)

### Conclusion

Thus, the only rational numbers in the domain of the function \(y\) are \(-1\) and \(1\). Therefore, the number of rational numbers in the domain of the function is:

\[ \text{Answer: } 2 \]