Consider three points `P = (-sin (beta-alpha), -cos beta)`, `Q = (cos(beta-alpha), sin beta)`, and `R = ((cos (beta - alpha + theta), sin (beta - theta))`, where `0< alpha, beta, theta < pi/4` Then

Options

A. P lies on the line segment RQ
B. Q lies on the segment PR
C. R lies on the line segment PR
D. P,Q,R are non-collinear

Explanation

To solve the problem of determining the relationship between the points \( P \), \( Q \), and \( R \), we will follow these steps:

### Step 1: Define the Points
We have three points defined as follows:
- \( P = (-\sin(\beta - \alpha), -\cos(\beta)) \)
- \( Q = (\cos(\beta - \alpha), \sin(\beta)) \)
- \( R = (\cos(\beta - \alpha + \theta), \sin(\beta - \theta)) \)

### Step 2: Express the Coordinates
Let’s express the coordinates of each point in terms of \( x \) and \( y \):
- For point \( P \):
- \( x_1 = -\sin(\beta - \alpha) \)
- \( y_1 = -\cos(\beta) \)

- For point \( Q \):
- \( x_2 = \cos(\beta - \alpha) \)
- \( y_2 = \sin(\beta) \)

- For point \( R \):
- \( x_3 = \cos(\beta - \alpha + \theta) \)
- \( y_3 = \sin(\beta - \theta) \)

### Step 3: Expand the Coordinates of Point R
Using the angle addition formulas, we can expand the coordinates of point \( R \):
- For \( x_3 \):
\[ x_3 = \cos(\beta - \alpha + \theta) = \cos(\beta - \alpha)\cos(\theta) - \sin(\beta - \alpha)\sin(\theta) \]
Substituting \( x_2 \) and \( y_1 \):
\[ x_3 = x_2 \cos(\theta) - (-y_1) \sin(\theta) = x_2 \cos(\theta) + y_1 \sin(\theta) \]

- For \( y_3 \):
\[ y_3 = \sin(\beta - \theta) = \sin(\beta)\cos(\theta) - \cos(\beta)\sin(\theta) \]
Substituting \( y_2 \) and \( y_1 \):
\[ y_3 = y_2 \cos(\theta) - (-y_1) \sin(\theta) = y_2 \cos(\theta) + y_1 \sin(\theta) \]

### Step 4: Check for Collinearity
To determine if points \( P \), \( Q \), and \( R \) are collinear, we can check if the area formed by these points is zero. The area can be calculated using the determinant:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

### Step 5: Substitute the Coordinates
Substituting the coordinates into the area formula:
\[ \text{Area} = \frac{1}{2} \left| (-\sin(\beta - \alpha))(\sin(\beta) - (y_2 \cos(\theta) + y_1 \sin(\theta))) + (\cos(\beta - \alpha))((y_2 \cos(\theta) + y_1 \sin(\theta)) - (-\cos(\beta))) + (\cos(\beta - \alpha + \theta)((-\cos(\beta)) - \sin(\beta))) \right| \]

### Step 6: Analyze the Result
If the area is non-zero, then the points are non-collinear. Given the constraints \( 0 < \alpha, \beta, \theta < \frac{\pi}{4} \), we can conclude that the points \( P \), \( Q \), and \( R \) do not lie on the same line.

### Conclusion
Thus, the final conclusion is that points \( P \), \( Q \), and \( R \) are non-collinear.