If A= $\frac{\pi}{6} and B= \frac{\pi}{3}$, then 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?

Question

If A= $\frac{\pi}{6} and B= \frac{\pi}{3}$, then 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?

Accepted Answer

To determine why Option 3 is the correct answer, we need to evaluate both statements I and II using the given values of $ A = \frac{\pi}{6} $ and $ B = \frac{\pi}{3} $.

**Statement I: $ \sin A + \sin B = \cos A + \cos B $**

First, calculate each trigonometric function:

- $ \sin A = \sin \frac{\pi}{6} = \frac{1}{2} $
- $ \sin B = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} $
- $ \cos A = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} $
- $ \cos B = \cos \frac{\pi}{3} = \frac{1}{2} $

Now, substitute these values into the equation:

$$
\sin A + \sin B = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2}
$$

$$
\cos A + \cos B = \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{\sqrt{3} + 1}{2}
$$

Since $ \frac{1 + \sqrt{3}}{2} = \frac{\sqrt{3} + 1}{2} $, Statement I is correct.

**Statement II: $ \tan A + \tan B = \cot A + \cot B $**

Calculate each trigonometric function:

- $ \tan A = \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} $
- $ \tan B = \tan \frac{\pi}{3} = \sqrt{3} $
- $ \cot A = \cot \frac{\pi}{6} = \sqrt{3} $
- $ \cot B = \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} $

Now, substitute these values into the equation:

$$
\tan A + \tan B = \frac{1}{\sqrt{3}} + \sqrt{3} = \frac{1}{\sqrt{3}} + \frac{3}{\sqrt{3}} = \frac{4}{\sqrt{3}}
$$

$$
\cot A + \cot B = \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}}
$$

Since $ \frac{4}{\sqrt{3}} = \frac{4}{\sqrt{3}} $, Statement II is also correct.

**Conclusion:**

Both statements I and II are correct, which supports Option 3 as the correct answer: "Both I and II."

**Explanation of Incorrect Options:**

- **Option 1 (Only I):** This is incorrect because both statements I and II are correct, not just I.
- **Option 2 (Only II):** This is incorrect because both statements I and II are correct, not just II.
- **Option 4 (Neither I nor II):** This is incorrect because both statements I and II are correct.

Thus, Option 3 is the correct answer, as both statements are indeed correct.