Let sinA+ sinB = a and cosA+ cosB = b, then the value

Solution

We are given:

• sin A + sin B = a
• cos A + cos B = b

We are to find the value of: a² + b² = (sin A + sin B)² + (cos A + cos B)² Let’s expand both terms: a² = (sin A + sin B)² = sin²A + sin²B + 2 sin A sin B b² = (cos A + cos B)² = cos²A + cos²B + 2 cos A cos B Now add them: a² + b² = (sin²A + sin²B + 2 sin A sin B) + (cos²A + cos²B + 2 cos A cos B) Group terms: = (sin²A + cos²A) + (sin²B + cos²B) + 2(sin A sin B + cos A cos B) Now use identities:

• sin²A + cos²A = 1
• sin²B + cos²B = 1
• sin A sin B + cos A cos B = cos(A – B)

So: a² + b² = 1 + 1 + 2 cos(A – B) ⇒ a² + b² = 2 + 2 cos(A – B)