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    Question

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

    of  a²  + b²  = ?
    A 4cos²((A–B)/2) Correct Answer Incorrect Answer
    B 2 - 2cos(A–B) Correct Answer Incorrect Answer
    C 4cos²((A+B)/2) Correct Answer Incorrect Answer
    D 2 + 2cos(A–B) Correct Answer Incorrect Answer

    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)

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