Find the solutions of the equation `cos ( cos^(-1) x) = cosec ( cosec^(-1)x).`

Answer

To solve the equation \( \cos(\cos^{-1} x) = \csc(\csc^{-1} x) \), we can follow these steps:

### Step 1: Understand the Functions
We know that:
- \( \cos(\cos^{-1} x) = x \) for \( x \) in the domain of \( \cos^{-1} \), which is \( [-1, 1] \).
- \( \csc(\csc^{-1} x) = x \) for \( x \) in the domain of \( \csc^{-1} \), which is \( (-\infty, -1] \cup [1, \infty) \).

### Step 2: Set Up the Equation
The equation becomes:
\[ x = x \]
This is trivially true, but we must consider the domains of the functions involved.

### Step 3: Identify the Domains
- The domain of \( \cos^{-1} x \) is \( x \in [-1, 1] \).
- The domain of \( \csc^{-1} x \) is \( x \in (-\infty, -1] \cup [1, \infty) \).

### Step 4: Find the Intersection of Domains
The intersection of the domains is where both functions are defined:
- From \( \cos^{-1} x \): \( x \) can be \( [-1, 1] \).
- From \( \csc^{-1} x \): \( x \) can be \( (-\infty, -1] \cup [1, \infty) \).

The only values that satisfy both domains are \( x = -1 \) and \( x = 1 \).

### Step 5: Conclusion
Thus, the solutions to the equation \( \cos(\cos^{-1} x) = \csc(\csc^{-1} x) \) are:
\[ x = -1 \quad \text{and} \quad x = 1 \]