Maths-
General
Easy

Question

In straight triangle A B C comma cot invisible function application A over 2 plus cot invisible function application B over 2 plus cot invisible function application C over 2 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 have to find we will first use thec o t A over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root comma space c o t space B over 2 equals square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root comma space c o t C over 2 equals square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root and further solve the equation to get the required value.

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


    space space space c o t space A over 2 plus c o t space B over 2 plus c o t space C over 2
equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root plus square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root
equals square root of fraction numerator s over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root left curly bracket left parenthesis s minus a right parenthesis plus left parenthesis s minus b right parenthesis plus left parenthesis s minus c right parenthesis right curly bracket
m u l t i p l y i n g space t h e space n u m e r a t o r space a n d space d e n o m i n a t o r space w i t h space square root of s

equals fraction numerator s over denominator square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root end fraction left curly bracket 3 s minus left parenthesis a plus b plus c right parenthesis right curly bracket
equals fraction numerator s left parenthesis 3 s minus left parenthesis a plus b plus c right parenthesis right parenthesis over denominator increment end fraction
equals fraction numerator begin display style fraction numerator left parenthesis a plus b plus c right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator 3 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style minus begin display style fraction numerator 2 open parentheses a plus b plus c close parentheses over denominator 2 end fraction end style close square brackets over denominator increment end fraction
equals fraction numerator open parentheses a plus b plus c close parentheses squared over denominator 4 increment end fraction

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