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AP EAMCET202120 Aug 2021Evening ShiftMathematicsTrigonometric Ratios & IdentitiesActual

If A + B + C = 3 π 2 , then cos 2 A + cos 2 B + cos 2 C =

Options

  1. A1 - 4 sin A sin B sin C
  2. B1 + 4 sin A sin B sin C
  3. C1 - 2 sin A sin B sin C
  4. D1 + 2 sin A sin B sin C

Correct answer

A. 1 - 4 sin A sin B sin C

Step-by-step solution

cos 2 A + cos 2 B + cos 2 C = 2 cos A + B cos A - B + cos 2 C ∵ cos C + cos D = 2 cos C + D 2 cos C - D 2 = 2 cos 3 π 2 - C cos A - B + 1 - 2 sin 2 C ∵ A + B + C = 3 π 2   &   cos 2 θ = 1 - 2 sin 2 θ = 1 - 2 sin C cos A - B - 2 sin 2 C = 1 - 2 sin C cos A - B + sin 3 π 2 - A + B ∵ A + B + C = 3 π 2 = 1 - 2 sin C cos A - B - cos A + B = 1 - 4 sin A sin B sin C ∵ cos C - cos D = 2 sin C + D 2 sin D - C 2

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