Question

In a triangle ABC, if \( (r_1 + r_2) \csc^2 \frac{C}{2} = \)

• A. \( 2R \cot^2 \frac{C}{2} \)
• B. \( 4R \tan^2 \frac{C}{2} \)
• C. \( 4R \cot^2 \frac{C}{2} \)
• D. \( 2R \tan^2 \frac{C}{2} \)

Hint:

In trigonometric identities in triangle geometry, certain standard formulas are derived from the relationship between the radius, angles, and sides of the triangle. Always be familiar with these relations for quick calculations.

Answer 1

The Correct Option is C

Step 1: Understand the given expression.
We are given the expression: \[ (r_1 + r_2) \csc^2 C. \] Here, \( r_1 \) and \( r_2 \) are the inradii, and \( C \) is an angle of the triangle. 

Step 2: Apply the standard result from triangle geometry.
From standard results in triangle geometry, we know: \[ (r_1 + r_2) \csc^2 C = 4R \cot^2 C, \] where \( R \) is the circumradius of the triangle, and \( C \) is the angle opposite side \( c \). 

Conclusion:
Thus, the correct expression is: \[ 4R \cot^2 C. \]