Find maximum value of `(1)/(sec theta), 0^(@) le theta le 90^(@)` .

Answer

To find the maximum value of \( \frac{1}{\sec \theta} \) for \( 0^\circ \leq \theta \leq 90^\circ \), we can follow these steps:

### Step 1: Rewrite the expression
We know that the secant function is the reciprocal of the cosine function. Therefore, we can rewrite the expression as:
\[ \frac{1}{\sec \theta} = \cos \theta \]

### Step 2: Analyze the cosine function
The cosine function, \( \cos \theta \), varies between 1 and 0 as \( \theta \) changes from \( 0^\circ \) to \( 90^\circ \):
- At \( \theta = 0^\circ \), \( \cos 0^\circ = 1 \)
- At \( \theta = 90^\circ \), \( \cos 90^\circ = 0 \)

### Step 3: Determine the maximum value
Since \( \cos \theta \) decreases from 1 to 0 in the interval \( [0^\circ, 90^\circ] \), the maximum value of \( \cos \theta \) occurs at \( \theta = 0^\circ \):
\[ \text{Maximum value of } \cos \theta = \cos 0^\circ = 1 \]

### Step 4: Conclusion
Thus, the maximum value of \( \frac{1}{\sec \theta} \) in the given range is:
\[ \boxed{1} \]