Number of integral values of `a` for which the equation `cos^2x-sinx+a=0` has roots when `x in (0,pi/2)` is____________

Explanation

To solve the equation \( \cos^2 x - \sin x + a = 0 \) for the number of integral values of \( a \) such that the equation has roots in the interval \( (0, \frac{\pi}{2}) \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cos^2 x - \sin x + a = 0 \]
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite the equation as:
\[ 1 - \sin^2 x - \sin x + a = 0 \]
This simplifies to:
\[ -\sin^2 x - \sin x + (a + 1) = 0 \]
or
\[ \sin^2 x + \sin x + (1 - a) = 0 \]

### Step 2: Analyze the quadratic in \( \sin x \)
Let \( y = \sin x \). The equation becomes:
\[ y^2 + y + (1 - a) = 0 \]
This is a quadratic equation in \( y \).

### Step 3: Determine the conditions for real roots
For the quadratic equation \( y^2 + y + (1 - a) = 0 \) to have real roots, the discriminant must be non-negative:
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (1 - a) \geq 0 \]
Calculating the discriminant:
\[ D = 1 - 4(1 - a) = 1 - 4 + 4a = 4a - 3 \]
Setting the discriminant greater than or equal to zero gives:
\[ 4a - 3 \geq 0 \implies 4a \geq 3 \implies a \geq \frac{3}{4} \]

### Step 4: Find the range of \( y \)
Since \( y = \sin x \) and \( x \) is in the interval \( (0, \frac{\pi}{2}) \), we have:
\[ 0 < y < 1 \]

### Step 5: Determine the conditions for roots to be in the interval
For the roots of the quadratic \( y^2 + y + (1 - a) = 0 \) to lie within the interval \( (0, 1) \), we need to check the values of \( y \) at the endpoints:
1. The sum of the roots \( y_1 + y_2 = -b/a = -1 \) (which is negative).
2. The product of the roots \( y_1 y_2 = c/a = 1 - a \).

For both roots to be positive:
\[ 1 - a > 0 \implies a < 1 \]

### Step 6: Combine inequalities
From the conditions derived:
\[ \frac{3}{4} \leq a < 1 \]
The integral values of \( a \) in this range are:
- The only integer satisfying this is \( a = 0 \).

### Conclusion
Thus, the number of integral values of \( a \) for which the equation has roots in the interval \( (0, \frac{\pi}{2}) \) is:
\[ \text{Number of integral values of } a = 1 \]