The equation `cos ^(2) x - sin x+lamda = 0, x in (0, pi//2)` has roots then value(s) of `lamda` can be equal to :

Options

A. 0
B. `1//2`
C. `-1`
D. `1//2`

Answer

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

### Step 1: Rewrite the equation
Start with the given equation:
\[ \cos^2 x - \sin x + \lambda = 0 \]
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite the equation as:
\[ 1 - \sin^2 x - \sin x + \lambda = 0 \]

### Step 2: Rearrange the equation
Rearranging gives:
\[ -\sin^2 x - \sin x + (1 + \lambda) = 0 \]
Multiplying through by -1, we have:
\[ \sin^2 x + \sin x - (1 + \lambda) = 0 \]

### Step 3: Identify the quadratic form
This is a quadratic equation in terms of \( \sin x \). Let \( y = \sin x \). The equation becomes:
\[ y^2 + y - (1 + \lambda) = 0 \]

### Step 4: Apply the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 1, c = -(1 + \lambda) \). Thus, we have:
\[ y = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot -(1 + \lambda)}}{2 \cdot 1} \]
This simplifies to:
\[ y = \frac{-1 \pm \sqrt{1 + 4 + 4\lambda}}{2} \]
\[ y = \frac{-1 \pm \sqrt{5 + 4\lambda}}{2} \]

### Step 5: Determine conditions for roots
For the equation to have real roots, the discriminant must be non-negative:
\[ 5 + 4\lambda \geq 0 \]
This leads to:
\[ 4\lambda \geq -5 \quad \Rightarrow \quad \lambda \geq -\frac{5}{4} \]

### Step 6: Determine the range of \( y \)
Since \( y = \sin x \) and \( x \) is in the interval \( (0, \frac{\pi}{2}) \), \( y \) must be in the range \( (0, 1) \). Therefore, we need:
\[ 0 < \frac{-1 + \sqrt{5 + 4\lambda}}{2} < 1 \]

### Step 7: Solve the inequalities
1. For the lower bound:
\[ -1 + \sqrt{5 + 4\lambda} > 0 \quad \Rightarrow \quad \sqrt{5 + 4\lambda} > 1 \]
Squaring both sides gives:
\[ 5 + 4\lambda > 1 \quad \Rightarrow \quad 4\lambda > -4 \quad \Rightarrow \quad \lambda > -1 \]

2. For the upper bound:
\[ -1 + \sqrt{5 + 4\lambda} < 2 \quad \Rightarrow \quad \sqrt{5 + 4\lambda} < 3 \]
Squaring both sides gives:
\[ 5 + 4\lambda < 9 \quad \Rightarrow \quad 4\lambda < 4 \quad \Rightarrow \quad \lambda < 1 \]

### Conclusion
Combining the inequalities, we find:
\[ -1 < \lambda < 1 \]
Thus, the values of \( \lambda \) for which the equation has roots in the interval \( (0, \frac{\pi}{2}) \) are:
\[ \lambda \in (-1, 1) \]