Trigonometry

Group terms: .
Use sum-to-product identity for : Thus: Factor: So, either: - Case 1: .
- Case 2: .
Thus, or .
General solution:

Let , so , . Then: Thus, . Consider: Let , so , . Then: Thus, . But: Since , consider : Since , , , and .
Thus, .

Step 1: Calculate the Semi-Perimeter
The semi-perimeter of the triangle is computed as: Step 2: Apply the Half-Angle Formula for Sine
The half-angle formula for sine is: Step 3: Substituting the Given Values
Substitute the known values into the formula:

Step 1: Apply the Cosine Rule
The cosine of the angles in a triangle can be expressed as: Step 2: Set Up the Summation
The desired summation is: Substituting the cosine formulas: Step 3: Simplify the Expression
Now, combine the terms: Simplifying further:

Step 1: Use the Identity
Step 2: Apply Given Values