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General
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Question

In straight triangle A B C comma fraction numerator a left parenthesis cos invisible function application B plus cos invisible function application C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction equals

  1. cos squared invisible function application A over 2
  2. sim squared invisible function application A over 2
  3. tan squared invisible function application A over 2
  4. cot squared invisible function application A over 2

Hint:

To simplify the given question we will first use the projection formula and then using other trigonometric identities we will simplify the question.

The correct answer is: sim squared invisible function application A over 2


    fraction numerator a left parenthesis cos B plus cos C right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator c minus b cos A plus b minus c space cos A over denominator 2 left parenthesis b plus c right parenthesis end fraction space space space space space space space open square brackets c equals a cos B plus b cos A rightwards double arrow b cos A equals c minus a cos B comma space b equals a cos C plus c space cos A rightwards double arrow c space cos space A equals b minus a cos C close square brackets
equals fraction numerator b plus c minus b cos A space minus c space cos A over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator b plus c minus cos A left parenthesis b plus c right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator left parenthesis b plus c right parenthesis left parenthesis 1 minus cos A right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction
equals fraction numerator 1 minus cos A over denominator 2 end fraction space space space space space space space space space space space space space space space space space space space space
equals sin squared A over 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets cos space 2 x equals 1 minus 2 sin squared x rightwards double arrow 2 sin squared x equals 1 minus 2 cos 2 x rightwards double arrow sin squared x equals fraction numerator 1 minus cos 2 x over denominator 2 end fraction close square brackets

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