Maths-
General
Easy

Question

In straight triangle A B C comma left parenthesis a plus b plus c right parenthesis open parentheses tan invisible function application A over 2 plus tan invisible function application B over 2 close parentheses equals

  1. 2 ccot invisible function application A over 2
  2. 2 c cot invisible function application B over 2
  3. 2 ccot invisible function application c over 2
  4. 2 c tan invisible function application c over 2

Hint:

To simplify the given equation we will use a+ b+ c = 2s, tanA over 2=square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root and tanB over 2=square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root then using other trigonometric identities we will simplify the equation.

The correct answer is: 2 ccot invisible function application c over 2


    As, a+ b+ c = 2s, tanA over 2=square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root , tanB over 2=square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root
    we have, (a +b +c) (tanA over 2+tanB over 2)
    equals 2 s open curly brackets square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis s minus c right parenthesis left parenthesis s minus a right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets square root of fraction numerator left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis end fraction end root close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator square root of s minus b end root square root of s minus b end root plus square root of s minus a end root square root of s minus a end root over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator s minus b plus s minus a over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 square root of fraction numerator s squared left parenthesis s minus c right parenthesis over denominator s end fraction end root open curly brackets fraction numerator 2 s minus left parenthesis a plus b right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 square root of s left parenthesis s minus c right parenthesis end root open curly brackets fraction numerator c over denominator square root of s minus a end root square root of s minus b end root end fraction close curly brackets
equals 2 c. square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction end root
equals 2 c. c o t C 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 space space space open square brackets square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator square root of s minus a end root square root of s minus b end root end fraction end root equals c o t C over 2 close square brackets

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