The smallest positive value of x (in degrees) for whichcos `tanx=(cos5^@ cos 20^@ + cos 35^@ cos 50^@-sin 50^@ sin 20^@ - sin35^@ sin50^@)/(sin5^@ cos 20^@ - sin 35^@ cos 50^@ + cos 5^@ sin 20^@ - cos 35^@ sin 50^@)` is equal to

Options

A. `-(1)/(sqrt(3))`
B. `(1)/(sqrt(3))`
C. `-sqrt(3)`
D. `sqrt(3)`

Explanation

To solve the given problem, we need to simplify the expression on the right side of the equation and then find the smallest positive value of \( x \) such that \( \tan x = \text{expression} \).

### Step-by-step Solution:

1. **Rewrite the Expression**:
We have:
\[ \tan x = \frac{\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 50^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ}{\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ} \]

2. **Use Trigonometric Identities**:
- For the numerator:
- \( \cos A \cos B - \sin A \sin B = \cos(A + B) \)
- Thus, \( \cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ = \cos(5^\circ + 20^\circ) + \cos(35^\circ + 50^\circ) \)
- This simplifies to \( \cos 25^\circ + \cos 85^\circ \).

- For the second part of the numerator:
- \( -\sin A \sin B = -\sin(50^\circ + 20^\circ) \)
- Thus, we have:
\[ \cos 25^\circ + \cos 85^\circ - \sin 70^\circ \]

3. **Simplifying the Denominator**:
- For the denominator:
- \( \sin A \cos B + \cos A \sin B = \sin(A + B) \)
- Thus, \( \sin 5^\circ \cos 20^\circ + \cos 5^\circ \sin 20^\circ = \sin(5^\circ + 20^\circ) = \sin 25^\circ \).
- For the second part of the denominator:
- \( -\sin A \cos B + \cos A \sin B = \sin(A - B) \)
- Thus, we have:
\[ \sin 25^\circ - \sin 85^\circ \]

4. **Combine the Results**:
Now we can rewrite the expression:
\[ \tan x = \frac{\cos 25^\circ + \cos 85^\circ - \sin 70^\circ}{\sin 25^\circ - \sin 85^\circ} \]

5. **Using Further Identities**:
- We can simplify \( \cos 85^\circ \) to \( \sin 5^\circ \) and \( \sin 70^\circ \) to \( \cos 20^\circ \).
- The denominator can be simplified using the identity for sine subtraction.

6. **Final Simplification**:
After applying the identities and simplifying, we reach:
\[ \tan x = -\cot 30^\circ \]
Since \( \cot 30^\circ = \sqrt{3} \), we have:
\[ \tan x = -\sqrt{3} \]

7. **Finding the Smallest Positive Value of \( x \)**:
The angle \( x \) that satisfies \( \tan x = -\sqrt{3} \) is:
\[ x = 180^\circ - 60^\circ = 120^\circ \]
Therefore, the smallest positive value of \( x \) is:
\[ \boxed{120^\circ} \]