If `secA=cosecA" and "0^(@) le A le 90^(@)`, state the value of A.

Answer

To solve the equation \( \sec A = \csc A \) where \( 0^\circ \leq A \leq 90^\circ \), we can follow these steps:

### Step 1: Write the definitions of secant and cosecant
The secant and cosecant functions are defined as:
\[ \sec A = \frac{1}{\cos A} \quad \text{and} \quad \csc A = \frac{1}{\sin A} \]

### Step 2: Set the two definitions equal to each other
From the given equation \( \sec A = \csc A \), we can substitute the definitions:
\[ \frac{1}{\cos A} = \frac{1}{\sin A} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ \sin A = \cos A \]

### Step 4: Use the relationship between sine and cosine
The equation \( \sin A = \cos A \) implies that the angle \( A \) must be such that the sine and cosine values are equal. This occurs at:
\[ A = 45^\circ \]

### Step 5: Verify the solution
Since \( 0^\circ \leq A \leq 90^\circ \), we check if \( A = 45^\circ \) satisfies the original equation:
- Calculate \( \sec 45^\circ \):
\[ \sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]
- Calculate \( \csc 45^\circ \):
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]

Since both values are equal, \( \sec 45^\circ = \csc 45^\circ \).

### Conclusion
Thus, the value of \( A \) is:
\[ \boxed{45^\circ} \]