Consider equation `(x - sin alpha) (x-cos alpha) - 2 = 0` . Which of the following is /are true?

Options

A. If `0lt alpha lt (pi)/(4)`, then the equation has both roots in `(sin alpha, cos alpha)`
B. If `(pi)/(4) lt alpha (pi)/(2)`, then the equations has both roots in `(sin alpha, cos alpha oo)`
C. If `0lt alpha lt (pi)/(4)`, the one roots lies in `(-oo, sin alpha)` and the other in `(sin alpha, oo)`
D. If `(pi)/(4) lt alpha lt (pi)/(2)` then one root lies in `(-oo, cos alpha )` and the other is `(sin alpha, oo)`

Explanation

To solve the equation \((x - \sin \alpha)(x - \cos \alpha) - 2 = 0\), we will follow these steps:

### Step 1: Expand the equation
We start by expanding the left-hand side of the equation:
\[ (x - \sin \alpha)(x - \cos \alpha) = x^2 - (\sin \alpha + \cos \alpha)x + \sin \alpha \cos \alpha \]
Thus, the equation becomes:
\[ x^2 - (\sin \alpha + \cos \alpha)x + \sin \alpha \cos \alpha - 2 = 0 \]

### Step 2: Identify the function
Let \(f(x) = x^2 - (\sin \alpha + \cos \alpha)x + \sin \alpha \cos \alpha - 2\).

### Step 3: Evaluate \(f(\sin \alpha)\) and \(f(\cos \alpha)\)
Now we will evaluate \(f(\sin \alpha)\):
\[ f(\sin \alpha) = (\sin \alpha)^2 - (\sin \alpha + \cos \alpha)\sin \alpha + \sin \alpha \cos \alpha - 2 \]
This simplifies to:
\[ f(\sin \alpha) = \sin^2 \alpha - \sin^2 \alpha - \sin \alpha \cos \alpha + \sin \alpha \cos \alpha - 2 = -2 \]

Next, we evaluate \(f(\cos \alpha)\):
\[ f(\cos \alpha) = (\cos \alpha)^2 - (\sin \alpha + \cos \alpha)\cos \alpha + \sin \alpha \cos \alpha - 2 \]
This simplifies to:
\[ f(\cos \alpha) = \cos^2 \alpha - \sin \alpha \cos \alpha - \cos^2 \alpha + \sin \alpha \cos \alpha - 2 = -2 \]

### Step 4: Analyze the behavior of \(f(x)\)
Since \(f(\sin \alpha) = -2\) and \(f(\cos \alpha) = -2\), we can conclude that both points lie below the x-axis.

### Step 5: Determine the roots
The function \(f(x)\) is a quadratic function and opens upwards (since the coefficient of \(x^2\) is positive).

- For \(0 < \alpha < \frac{\pi}{4}\):
- Here, \(\cos \alpha > \sin \alpha\).
- The roots \(a\) and \(b\) must lie between \(-\infty\) and \(\sin \alpha\) for one root, and between \(\cos \alpha\) and \(\infty\) for the other root.

- For \(\frac{\pi}{4} < \alpha < \frac{\pi}{2}\):
- Here, \(\sin \alpha > \cos \alpha\).
- The roots \(a\) and \(b\) must lie between \(-\infty\) and \(\cos \alpha\) for one root, and between \(\sin \alpha\) and \(\infty\) for the other root.

### Conclusion
From the analysis, we can conclude that:
- For \(0 < \alpha < \frac{\pi}{4}\), one root lies in \((- \infty, \sin \alpha)\) and the other in \((\cos \alpha, \infty)\).
- For \(\frac{\pi}{4} < \alpha < \frac{\pi}{2}\), one root lies in \((- \infty, \cos \alpha)\) and the other in \((\sin \alpha, \infty)\).

Thus, options C and D are correct.