If `f(theta) = (1- sin 2theta + cos 2theta)/(2sin 2theta)` then value of` f(11^o). f(34^o)` is

Explanation

To solve the problem, we need to evaluate the function \( f(\theta) = \frac{1 - \sin 2\theta + \cos 2\theta}{2 \sin 2\theta} \) for \( \theta = 11^\circ \) and \( \theta = 34^\circ \), and then find the product \( f(11^\circ) \cdot f(34^\circ) \).

### Step-by-Step Solution:

1. **Rewrite the Function**:
We start with the function:
\[ f(\theta) = \frac{1 - \sin 2\theta + \cos 2\theta}{2 \sin 2\theta} \]

2. **Use Trigonometric Identities**:
Recall that:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
and
\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 \]
Thus, we can rewrite:
\[ f(\theta) = \frac{1 - 2 \sin \theta \cos \theta + (2 \cos^2 \theta - 1)}{2 \cdot 2 \sin \theta \cos \theta} \]
Simplifying the numerator:
\[ f(\theta) = \frac{2 \cos^2 \theta - 2 \sin \theta \cos \theta}{4 \sin \theta \cos \theta} \]
Factoring out the common terms:
\[ f(\theta) = \frac{2 \cos \theta (\cos \theta - \sin \theta)}{4 \sin \theta \cos \theta} = \frac{\cos \theta - \sin \theta}{2 \sin \theta} \]

3. **Evaluate \( f(11^\circ) \)**:
\[ f(11^\circ) = \frac{\cos 11^\circ - \sin 11^\circ}{2 \sin 11^\circ} \]

4. **Evaluate \( f(34^\circ) \)**:
\[ f(34^\circ) = \frac{\cos 34^\circ - \sin 34^\circ}{2 \sin 34^\circ} \]

5. **Find the Product \( f(11^\circ) \cdot f(34^\circ) \)**:
\[ f(11^\circ) \cdot f(34^\circ = \left( \frac{\cos 11^\circ - \sin 11^\circ}{2 \sin 11^\circ} \right) \cdot \left( \frac{\cos 34^\circ - \sin 34^\circ}{2 \sin 34^\circ} \right) \]
This simplifies to:
\[ = \frac{(\cos 11^\circ - \sin 11^\circ)(\cos 34^\circ - \sin 34^\circ)}{4 \sin 11^\circ \sin 34^\circ} \]

6. **Use the Angle Difference Identity**:
We can use the identity for \( \tan(45^\circ - x) \):
\[ \tan(45^\circ - 11^\circ) = \frac{1 - \tan 11^\circ}{1 + \tan 11^\circ} \]
Thus, \( \tan(34^\circ) = \tan(45^\circ - 11^\circ) \).

7. **Final Calculation**:
After substituting and simplifying, we find that:
\[ f(11^\circ) \cdot f(34^\circ) = \frac{1}{2} \]

### Conclusion:
The value of \( f(11^\circ) \cdot f(34^\circ) \) is \( \frac{1}{2} \).