Find the value of x for which `sin^(-1) (cos^(-1) x) lt 1 and cos^(-1) (cos^(-1) x) lt 1`

Explanation

To solve the problem, we need to find the value of \( x \) for which the following two inequalities hold:

1. \( \sin^{-1}(\cos^{-1}(x)) < 1 \)
2. \( \cos^{-1}(\cos^{-1}(x)) < 1 \)

### Step 1: Analyze the first inequality
We start with the first inequality:
\[ \sin^{-1}(\cos^{-1}(x)) < 1 \]
The range of \( \sin^{-1}(y) \) is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). Since \( \sin^{-1}(y) < 1 \) implies \( y < \sin(1) \), we have:
\[ \cos^{-1}(x) < \sin(1) \]

### Step 2: Solve for \( x \) from the first inequality
Next, we apply the cosine function to both sides. Since \( \cos^{-1}(x) \) is a decreasing function, we can write:
\[ x > \cos(\sin(1)) \]

### Step 3: Analyze the second inequality
Now, we consider the second inequality:
\[ \cos^{-1}(\cos^{-1}(x)) < 1 \]
The range of \( \cos^{-1}(y) \) is \( [0, \pi] \). Thus, \( \cos^{-1}(x) < 1 \) implies:
\[ x > \cos(1) \]

### Step 4: Combine the inequalities
Now we have two inequalities:
1. \( x > \cos(\sin(1)) \)
2. \( x > \cos(1) \)

To find the solution set for \( x \), we need to take the maximum of the two lower bounds:
\[ x > \max(\cos(\sin(1)), \cos(1)) \]

### Step 5: Determine the final range for \( x \)
Finally, we also know that \( x \) must be within the range of the cosine function, which is:
\[ x \leq 1 \]

Thus, the final solution can be summarized as:
\[ \max(\cos(\sin(1)), \cos(1)) < x \leq 1 \]