Find the maximum value of `(sec^(-1) x) (cosec^(-1) x), x ge 1`

Explanation

To find the maximum value of the expression \( (sec^{-1} x) (cosec^{-1} x) \) for \( x \geq 1 \), we can follow these steps:

### Step 1: Define the variables
Let \( a = sec^{-1} x \) and \( b = cosec^{-1} x \). We need to find the maximum value of the product \( ab \).

### Step 2: Use the property of means
We can use the property that the arithmetic mean is always greater than or equal to the geometric mean. According to this property:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
This implies:
\[ a + b \geq 2\sqrt{ab} \]

### Step 3: Find \( a + b \)
From trigonometric identities, we know that:
\[ sec^{-1} x + cosec^{-1} x = \frac{\pi}{2} \]
Thus,
\[ a + b = \frac{\pi}{2} \]

### Step 4: Substitute into the inequality
Substituting \( a + b \) into the inequality gives:
\[ \frac{\pi}{2} \geq 2\sqrt{ab} \]

### Step 5: Square both sides
Squaring both sides of the inequality:
\[ \left(\frac{\pi}{2}\right)^2 \geq 4ab \]
This simplifies to:
\[ \frac{\pi^2}{4} \geq 4ab \]

### Step 6: Solve for \( ab \)
Dividing both sides by 4:
\[ \frac{\pi^2}{16} \geq ab \]
Thus, we have:
\[ ab \leq \frac{\pi^2}{16} \]

### Conclusion
The maximum value of the product \( (sec^{-1} x)(cosec^{-1} x) \) for \( x \geq 1 \) is:
\[ \frac{\pi^2}{16} \]