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Maths-

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Question

In $straight triangle A B C comma cot invisible function application A over 2 plus cot invisible function application B over 2 plus cot invisible function application C over 2 equals$

$\frac{a + b + c}{4 Δ}$
$\frac{a^{2} + b^{2} + c^{2}}{4 Δ}$
$\frac{(a + b + c)^{2}}{4 Δ}$
$\frac{(a + b + c)^{2}}{2 Δ}$

hint Hint:

In this question we have to find we will first use the $c o t A over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end fraction end root comma space c o t space B over 2 equals square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis end fraction end root comma space c o t C over 2 equals square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end fraction end root$ 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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