Solves `cos^(-1) x lt 2`

Options

A. `x in (cos2, 2)`
B. `x in (0, 1)`
C. `x in (-1, 1)`
D. `x in (cos2, 1)`

Explanation

To solve the inequality \( \cos^{-1}(x) < 2 \), we will follow these steps:

### Step 1: Understand the range of the inverse cosine function
The range of the function \( \cos^{-1}(x) \) is from \( 0 \) to \( \pi \). Therefore, we can express this as:
\[ 0 \leq \cos^{-1}(x) \leq \pi \]

### Step 2: Set up the inequality
Given the inequality \( \cos^{-1}(x) < 2 \), we can combine this with the range of \( \cos^{-1}(x) \):
\[ 0 \leq \cos^{-1}(x) < 2 \]

### Step 3: Apply the cosine function
Since the cosine function is decreasing in the range of \( [0, \pi] \), we can apply the cosine function to all parts of the inequality:
\[ \cos(2) < x \leq \cos(0) \]

### Step 4: Evaluate the cosine values
We know that:
\[ \cos(0) = 1 \]
and we need to find \( \cos(2) \). The value of \( \cos(2) \) can be calculated using a calculator or trigonometric tables. For this solution, we will denote \( \cos(2) \) as it is.

### Step 5: Write the final solution
Now we can express the solution in interval notation:
\[ \cos(2) < x \leq 1 \]

### Conclusion
Thus, the solution to the inequality \( \cos^{-1}(x) < 2 \) is:
\[ x \in (\cos(2), 1] \]