The triangle OAB is right angled where points O,A,B are (0,0) `(cos theta, sin theta)` and `(cos phi sin phi)` respectively , then `theta` and `phi` are connected by the relation

Options

A. `sin((theta-phi)/2)=1/(sqrt(2))`
B. `cos ((theta-phi)/2)=1/(sqrt(2))`
C. `cos ((theta-phi)/2)=-1/(sqrt(2))`
D. `sin((theta-phi)/2)=-1/(sqrt(2))`

Answer

To solve the problem, we need to find the relationship between the angles \( \theta \) and \( \phi \) given the coordinates of points O, A, and B in the triangle OAB.

### Step-by-Step Solution:

1. **Identify the Points**:
- Point O is at the origin (0, 0).
- Point A is at \( (\cos \theta, \sin \theta) \).
- Point B is at \( (\cos \phi, \sin \phi) \).

2. **Calculate the Lengths OA and OB**:
- The length \( OA \) can be calculated using the distance formula:
\[ OA = \sqrt{(\cos \theta - 0)^2 + (\sin \theta - 0)^2} = \sqrt{\cos^2 \theta + \sin^2 \theta} = 1 \]
- Similarly, for \( OB \):
\[ OB = \sqrt{(\cos \phi - 0)^2 + (\sin \phi - 0)^2} = \sqrt{\cos^2 \phi + \sin^2 \phi} = 1 \]

3. **Use the Pythagorean Theorem**:
- In triangle OAB, since it is a right triangle, we can use the Pythagorean theorem:
\[ AB^2 = OA^2 + OB^2 \]
- Substituting the values we found:
\[ AB^2 = 1^2 + 1^2 = 2 \]

4. **Calculate the Length AB**:
- The length \( AB \) can also be calculated as:
\[ AB = \sqrt{(\cos \theta - \cos \phi)^2 + (\sin \theta - \sin \phi)^2} \]
- Squaring both sides gives:
\[ AB^2 = (\cos \theta - \cos \phi)^2 + (\sin \theta - \sin \phi)^2 \]

5. **Set the Two Expressions for AB^2 Equal**:
- From the previous steps, we have:
\[ 2 = (\cos \theta - \cos \phi)^2 + (\sin \theta - \sin \phi)^2 \]

6. **Expand the Right Side**:
- Expanding the right side:
\[ (\cos \theta - \cos \phi)^2 + (\sin \theta - \sin \phi)^2 = (\cos^2 \theta - 2\cos \theta \cos \phi + \cos^2 \phi) + (\sin^2 \theta - 2\sin \theta \sin \phi + \sin^2 \phi) \]
- Using the identity \( \cos^2 x + \sin^2 x = 1 \):
\[ = 1 - 2\cos \theta \cos \phi + 1 - 2\sin \theta \sin \phi = 2 - 2(\cos \theta \cos \phi + \sin \theta \sin \phi) \]

7. **Use the Cosine Addition Formula**:
- The expression \( \cos \theta \cos \phi + \sin \theta \sin \phi \) can be rewritten using the cosine of the difference of angles:
\[ \cos(\theta - \phi) \]
- Therefore, we have:
\[ 2 = 2 - 2\cos(\theta - \phi) \]

8. **Simplify the Equation**:
- This simplifies to:
\[ 0 = -2\cos(\theta - \phi) \]
- Thus:
\[ \cos(\theta - \phi) = 0 \]

9. **Find the Relationship Between Angles**:
- The cosine of an angle is zero when the angle is \( \frac{\pi}{2} + n\pi \) for any integer \( n \). Therefore, we can conclude:
\[ \theta - \phi = \frac{\pi}{2} + n\pi \]

### Final Relation:
The angles \( \theta \) and \( \phi \) are connected by the relation:
\[ \theta - \phi = \frac{\pi}{2} + n\pi \]