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Question

In straight triangle A B C comma cot A plus cot invisible function application B plus cot invisible function application C equals

  1. fraction numerator a plus b plus c over denominator 4 straight capital delta end fraction
  2. fraction numerator a squared plus b squared plus c squared over denominator 4 straight capital delta end fraction
  3. fraction numerator left parenthesis a plus b plus c right parenthesis squared over denominator 4 straight capital delta end fraction
  4. fraction numerator left parenthesis a plus b plus c right parenthesis squared over denominator 2 straight capital delta end fraction

Hint:

In this question we will first covert cot into sin and cos form. Then we will use cosine formula and area of triangle open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses. Later we will simplify the equation to get the required value.

The correct answer is: fraction numerator a squared plus b squared plus c squared over denominator 4 straight capital delta end fraction


    space space space space c o t space A plus c o t space B plus space c o t space C space
equals fraction numerator cos space A over denominator sin A end fraction plus fraction numerator cos space B space over denominator sin space B space end fraction plus fraction numerator cos space C over denominator sin space C end fraction
equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction. fraction numerator b c over denominator 2 increment end fraction plus fraction numerator c squared plus a squared minus b squared over denominator 2 c a end fraction. fraction numerator c a over denominator 2 increment end fraction plus fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction. fraction numerator a b over denominator 2 increment end fraction space space space space space open square brackets u sin g space cos i n e space f o r m u l a space a n d space a r e a space o f space t r i a n g l e space open parentheses increment equals fraction numerator a b space sin C over denominator 2 end fraction equals fraction numerator b c space sin space A over denominator 2 end fraction equals fraction numerator c a space sin space B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator b squared plus c squared minus a squared plus c squared plus a squared minus b squared plus a squared plus b squared minus c squared over denominator 4 increment end fraction
equals fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction

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