If `A=pi/6 and B=pi/3`, ten consider the following statements:
I. sin A + sin B = cos A + cos B
II. Tan A + tan B = cot A + cot B
Which of the above statements is//are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To solve the problem, we need to evaluate the two statements given the angles \( A = \frac{\pi}{6} \) (which is 30 degrees) and \( B = \frac{\pi}{3} \) (which is 60 degrees).

### Step 1: Evaluate Statement I
Statement I: \( \sin A + \sin B = \cos A + \cos B \)

1. **Calculate \( \sin A \) and \( \sin B \)**:
- \( \sin A = \sin\left(\frac{\pi}{6}\right) = \sin(30^\circ) = \frac{1}{2} \)
- \( \sin B = \sin\left(\frac{\pi}{3}\right) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \)

2. **Sum \( \sin A + \sin B \)**:
\[ \sin A + \sin B = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2} \]

3. **Calculate \( \cos A \) and \( \cos B \)**:
- \( \cos A = \cos\left(\frac{\pi}{6}\right) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
- \( \cos B = \cos\left(\frac{\pi}{3}\right) = \cos(60^\circ) = \frac{1}{2} \)

4. **Sum \( \cos A + \cos B \)**:
\[ \cos A + \cos B = \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{1 + \sqrt{3}}{2} \]

5. **Compare the sums**:
\[ \sin A + \sin B = \cos A + \cos B \]
Thus, Statement I is **true**.

### Step 2: Evaluate Statement II
Statement II: \( \tan A + \tan B = \cot A + \cot B \)

1. **Calculate \( \tan A \) and \( \tan B \)**:
- \( \tan A = \tan\left(\frac{\pi}{6}\right) = \tan(30^\circ) = \frac{1}{\sqrt{3}} \)
- \( \tan B = \tan\left(\frac{\pi}{3}\right) = \tan(60^\circ) = \sqrt{3} \)

2. **Sum \( \tan A + \tan B \)**:
\[ \tan A + \tan B = \frac{1}{\sqrt{3}} + \sqrt{3} \]

3. **Calculate \( \cot A \) and \( \cot B \)**:
- \( \cot A = \cot\left(\frac{\pi}{6}\right) = \cot(30^\circ) = \sqrt{3} \)
- \( \cot B = \cot\left(\frac{\pi}{3}\right) = \cot(60^\circ) = \frac{1}{\sqrt{3}} \)

4. **Sum \( \cot A + \cot B \)**:
\[ \cot A + \cot B = \sqrt{3} + \frac{1}{\sqrt{3}} \]

5. **Compare the sums**:
\[ \tan A + \tan B = \cot A + \cot B \]
Both expressions simplify to the same value, thus Statement II is also **true**.

### Conclusion
Both statements are correct.

### Final Answer
Both statements I and II are correct.

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