Verifying Trigonometric Statements for Specific Angles

The problem asks us to verify the correctness of two trigonometric statements given specific values for angles A and B. The given values are $\text{A} = \frac{\pi}{6}$ and $\text{B} = \frac{\pi}{3}$.

First, let's convert the angles from radians to degrees for easier understanding of common trigonometric values:

• $\text{A} = \frac{\pi}{6} \text{ radians} = \frac{180^\circ}{6} = 30^\circ$
• $\text{B} = \frac{\pi}{3} \text{ radians} = \frac{180^\circ}{3} = 60^\circ$

Calculating Trigonometric Ratios for A and B

Now, let's find the values of the trigonometric ratios for A = $30^\circ$ and B = $60^\circ$:

• For $\text{A} = 30^\circ$:
  • $\text{sin A} = \text{sin } 30^\circ = \frac{1}{2}$
  • $\text{cos A} = \text{cos } 30^\circ = \frac{\sqrt{3}}{2}$
  • $\text{tan A} = \text{tan } 30^\circ = \frac{1}{\sqrt{3}}$
  • $\text{cot A} = \text{cot } 30^\circ = \sqrt{3}$
• For $\text{B} = 60^\circ$:
  • $\text{sin B} = \text{sin } 60^\circ = \frac{\sqrt{3}}{2}$
  • $\text{cos B} = \text{cos } 60^\circ = \frac{1}{2}$
  • $\text{tan B} = \text{tan } 60^\circ = \sqrt{3}$
  • $\text{cot B} = \text{cot } 60^\circ = \frac{1}{\sqrt{3}}$

Evaluating Statement I: sin A + sin B = cos A + cos B

Let's substitute the calculated values into Statement I:

Left Hand Side (LHS): $\text{sin A} + \text{sin B} = \text{sin } 30^\circ + \text{sin } 60^\circ = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2}$

Right Hand Side (RHS): $\text{cos A} + \text{cos B} = \text{cos } 30^\circ + \text{cos } 60^\circ = \frac{\sqrt{3}}{2} + \frac{1}{2} = \frac{\sqrt{3} + 1}{2}$

Comparing LHS and RHS:

$\frac{1 + \sqrt{3}}{2} = \frac{\sqrt{3} + 1}{2}$

The LHS is equal to the RHS. Therefore, Statement I is correct for the given values of A and B.

Evaluating Statement II: tan A + tan B = cot A + cot B

Now, let's substitute the calculated values into Statement II:

Left Hand Side (LHS): $\text{tan A} + \text{tan B} = \text{tan } 30^\circ + \text{tan } 60^\circ = \frac{1}{\sqrt{3}} + \sqrt{3}$

To add these, find a common denominator:

$\frac{1}{\sqrt{3}} + \sqrt{3} = \frac{1}{\sqrt{3}} + \frac{\sqrt{3} \times \sqrt{3}}{\sqrt{3}} = \frac{1}{\sqrt{3}} + \frac{3}{\sqrt{3}} = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}}$

Right Hand Side (RHS): $\text{cot A} + \text{cot B} = \text{cot } 30^\circ + \text{cot } 60^\circ = \sqrt{3} + \frac{1}{\sqrt{3}}$

Similarly, find a common denominator:

$\sqrt{3} + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} \times \sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{3 + 1}{\sqrt{3}} = \frac{4}{\sqrt{3}}$

Comparing LHS and RHS:

$\frac{4}{\sqrt{3}} = \frac{4}{\sqrt{3}}$

The LHS is equal to the RHS. Therefore, Statement II is correct for the given values of A and B.

Conclusion

Based on the evaluation of both statements using the given values $\text{A} = \frac{\pi}{6}$ and $\text{B} = \frac{\pi}{3}$, we found that both Statement I and Statement II are correct.