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Question

In $triangle A B C comma c o t invisible function application A divided by 2 plus c o t invisible function application B divided by 2 plus c o t invisible function application C divided by 2 equals$

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

hint Hint:

In this question we have to use the formula
$cot open parentheses A over 2 close parentheses equals space fraction numerator s left parenthesis s minus a right parenthesis over denominator increment end fraction cot open parentheses B over 2 close parentheses equals fraction numerator s left parenthesis s minus b right parenthesis over denominator increment end fraction cot open parentheses C over 2 close parentheses equals fraction numerator s left parenthesis s minus c right parenthesis over denominator increment end fraction$ where $increment equals square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root space space comma space s space equals space fraction numerator a plus b plus c over denominator 2 end fraction$

The correct answer is: $left parenthesis left parenthesis a plus b plus c right parenthesis squared right parenthesis divided by 4 capital delta$

In this question we have to find the value of $cot open parentheses A over 2 close parentheses plus cot open parentheses B over 2 close parentheses plus cot open parentheses C over 2 close parentheses$
We know that
$cot open parentheses A over 2 close parentheses equals space fraction numerator s left parenthesis s minus a right parenthesis over denominator increment end fraction cot open parentheses B over 2 close parentheses equals fraction numerator s left parenthesis s minus b right parenthesis over denominator increment end fraction cot open parentheses C over 2 close parentheses equals fraction numerator s left parenthesis s minus c right parenthesis over denominator increment end fraction$ where $increment equals square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root space space comma space s space equals space fraction numerator a plus b plus c over denominator 2 end fraction$
=> $cot open parentheses A over 2 close parentheses plus cot open parentheses B over 2 close parentheses plus cot open parentheses C over 2 close parentheses$
=> $fraction numerator s left parenthesis s minus a right parenthesis over denominator increment end fraction plus fraction numerator s left parenthesis s minus b right parenthesis over denominator increment end fraction plus fraction numerator s left parenthesis s minus c right parenthesis over denominator increment end fraction$
=> $fraction numerator s left parenthesis s minus a space plus space s minus b plus s minus c right parenthesis over denominator increment end fraction$
=> $fraction numerator s left parenthesis 3 s minus left parenthesis a plus b plus c right parenthesis right parenthesis over denominator increment end fraction$
=> $fraction numerator s left parenthesis 3 s minus 2 s right parenthesis over denominator increment end fraction$
=> $s squared over increment$
=> $left parenthesis begin display style fraction numerator a plus b plus c over denominator 2 end fraction end style right parenthesis squared over increment$
=> $fraction numerator left parenthesis begin display style a plus b plus c end style right parenthesis squared over denominator 4 increment end fraction$

Related Questions to study

In $capital delta A B C comma c o t invisible function application A plus c o t invisible function application B plus c o t invisible function application C equals$

In $capital delta A B C comma c o t invisible function application A plus c o t invisible function application B plus c o t invisible function application C equals$

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If 'O' is the Orthocentre of $triangle A B C$ then $O A equals$

If 'O' is the Orthocentre of $triangle A B C$ then $O A equals$

parallel

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parallel

In $triangle A B C comma s squared t a n invisible function application A divided by 2 t a n invisible function application B divided by 2 t a n invisible function application C divided by 2 equals$

In $triangle A B C comma s squared t a n invisible function application A divided by 2 t a n invisible function application B divided by 2 t a n invisible function application C divided by 2 equals$

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In $triangle A B C comma 4 R s sin invisible function application A divided by 2 space sin invisible function application B divided by 2 space sin invisible function application C divided by 2 equals$

In $A B C space comma space 4 R cos A divided by 2 space cos B divided by 2 space cos C divided by 2 equals$

In $A B C space comma space 4 R cos A divided by 2 space cos B divided by 2 space cos C divided by 2 equals$

parallel

In $triangle A B C comma sum a cos invisible function application A equals$

In $triangle A B C comma sum a cos invisible function application A equals$

In $capital delta A B C comma sum left parenthesis left parenthesis b squared minus c squared right parenthesis right parenthesis divided by a c o s invisible function application A equals$

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$L t left parenthesis x rightwards arrow pi divided by 6 right parenthesis left parenthesis left parenthesis 3 s i n x minus √ 3 c o s x right parenthesis divided by left parenthesis 6 x minus pi right parenthesis right parenthesis equals$

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parallel

If $L t left parenthesis x rightwards arrow 0 right parenthesis left parenthesis left parenthesis c o s invisible function application 4 x plus a c o s invisible function application 2 x plus b right parenthesis divided by x to the power of 4 right parenthesis$ is finite then the values of a,b are respectively

If $L t left parenthesis x rightwards arrow 0 right parenthesis left parenthesis left parenthesis c o s invisible function application 4 x plus a c o s invisible function application 2 x plus b right parenthesis divided by x to the power of 4 right parenthesis$ is finite then the values of a,b are respectively

If $f left parenthesis x right parenthesis equals x t a n left parenthesis negative 1 right parenthesis invisible function application x$ then $l i m subscript left parenthesis x rightwards arrow 1 right parenthesis left parenthesis f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis equals$

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parallel

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