Question

Assertion (A): If \( A = 10^\circ, B = 16^\circ, C = 19^\circ \), then: \[ \tan(2A) \tan(2B) + \tan(2B) \tan(2C) + \tan(2C) \tan(2A) = 1. \] Reason (R): If \( A + B + C = 180^\circ \), then: \[ \cot\left(\frac{A}{2}\right) + \cot\left(\frac{B}{2}\right) + \cot\left(\frac{C}{2}\right) = \cot\left(\frac{A}{2}\right) \cot\left(\frac{B}{2}\right) \cot\left(\frac{C}{2}\right). \]

• A. Both (A) and (R) are true and (R) is the correct explanation of (A).
• B. Both (A) and (R) are true and (R) is NOT correct explanation of (A).
• C. (A) is true, (R) is false.
• D. (A) is false, (R) is true.

Hint:

When working with trigonometric identities and angles in a triangle, check for sum or difference identities, and use appropriate angle values for verification.

Answer 1

The Correct Option is A

Step 1: Verifying the assertion. We are given the equation: \[ \tan(2A) \tan(2B) + \tan(2B) \tan(2C) + \tan(2C) \tan(2A) = 1, \] where \( A = 10^\circ, B = 16^\circ, C = 19^\circ \). After calculating the values of \( \tan(2A) \), \( \tan(2B) \), and \( \tan(2C) \), we find that the assertion holds true. 
Step 2: Verifying the reason. Reason (R) is a standard identity in trigonometry. Given that \( A + B + C = 180^\circ \), the identity is true, so Reason (R) is valid. 
Step 3: Conclusion. Thus, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation for Assertion (A).