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AP EAMCET20224 Jul 2022Evening ShiftMathematicsProperties of TrianglesActual

In a ∆ A B C , 2 ( b c cos A + c a cos B + a b cos C ) = ?

Options

  1. Aa 2 + b 2 - c 2
  2. Ba 2 + c 2 - b 2
  3. Cb 2 + c 2 - a 2
  4. Da 2 + b 2 + c 2

Correct answer

D. a 2 + b 2 + c 2

Step-by-step solution

Given that, the trigonometric expression is 2 ( b c cos A + c a cos B + a b cos C ) = 2 b c cos A + 2 c a cos B + 2 a b cos C We know that, cos A = b 2 + c 2 - a 2 2 b c . ⇒ 2 b c · b 2 + c 2 - a 2 2 b c + 2 c a · c 2 + a 2 - b 2 2 c a + 2 a b · a 2 + b 2 - c 2 2 a b = b 2 + c 2 - a 2 + c 2 + a 2 - b 2 + a 2 + b 2 - c 2 = a 2 + b 2 + c 2 Hence, the value of the given trigonometric expression is a 2 + b 2 + c 2 .

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