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Question

$I n space triangle A B C comma space i f space x equals$ $tan invisible function application left parenthesis left parenthesis B minus C right parenthesis divided by 2 right parenthesis tan invisible function application A divided by 2 comma y equals tan invisible function application left parenthesis left parenthesis C minus A right parenthesis divided by 2 right parenthesis tan invisible function application B divided by 2 comma z equals tan invisible function application left parenthesis left parenthesis A minus B right parenthesis divided by 2 right parenthesis tan invisible function application C divided by 2 space t space h space e n space x plus y plus z space i n space t e r m s space left parenthesis x comma y comma z space t e r m s space o n l y space right parenthesis space i s$

$x y z$
$2 x y z$
$- x y z$
$- x y z$

hint Hint:

In this question we have to use the formula to find the answer
$tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses space equals space fraction numerator b minus c over denominator b plus c end fraction cot open parentheses A over 2 close parentheses$
$tan open parentheses fraction numerator C minus A over denominator 2 end fraction close parentheses space equals space fraction numerator c minus a over denominator c plus a end fraction cot open parentheses B over 2 close parentheses$
$tan open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses space equals space fraction numerator a minus b over denominator a plus b end fraction cot open parentheses C over 2 close parentheses$

The correct answer is: $negative x y z$

In this question we are given
$x equals space tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses tan open parentheses A over 2 close parentheses comma y equals tan open parentheses fraction numerator C minus A over denominator 2 end fraction close parentheses tan open parentheses B over 2 close parentheses$ and $z equals space tan open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses tan open parentheses C over 2 close parentheses$
Step1: Using the formula from the solution of a triangle
$tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses space equals space fraction numerator b minus c over denominator b plus c end fraction cot open parentheses A over 2 close parentheses$
$tan open parentheses fraction numerator C minus A over denominator 2 end fraction close parentheses space equals space fraction numerator c minus a over denominator c plus a end fraction cot open parentheses B over 2 close parentheses$
$tan open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses space equals space fraction numerator a minus b over denominator a plus b end fraction cot open parentheses C over 2 close parentheses$
By putting these values in the above expression we get,
$x equals fraction numerator b minus c over denominator b plus c end fraction$ , $y equals fraction numerator c minus a over denominator c plus a end fraction$ and $z equals fraction numerator a minus b over denominator a plus b end fraction$
Step2: Finding the value of $x plus y plus z$ .
=> $x plus y plus z equals space fraction numerator b minus c over denominator b plus c end fraction plus fraction numerator c minus a over denominator c plus a end fraction plus fraction numerator a minus b over denominator a plus b end fraction$
=> $fraction numerator left parenthesis b minus c right parenthesis left parenthesis c plus a right parenthesis left parenthesis a plus b right parenthesis plus left parenthesis c minus a right parenthesis left parenthesis b plus c right parenthesis left parenthesis a plus b right parenthesis plus left parenthesis a minus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
By taking similar terms common
=> $fraction numerator left parenthesis b minus c right parenthesis left parenthesis c plus a right parenthesis left parenthesis a plus b right parenthesis plus left parenthesis c minus a right parenthesis left parenthesis b plus c right parenthesis left parenthesis a plus b right parenthesis plus left parenthesis a minus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $fraction numerator left parenthesis a plus b right parenthesis left parenthesis left parenthesis b minus c right parenthesis left parenthesis c plus a right parenthesis plus left parenthesis c minus a right parenthesis left parenthesis b plus c right parenthesis right parenthesis plus left parenthesis a minus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $fraction numerator left parenthesis a plus b right parenthesis left parenthesis left parenthesis b c plus a b minus c squared minus a c right parenthesis plus left parenthesis b c plus c squared minus a b minus a c right parenthesis right parenthesis plus left parenthesis a minus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $fraction numerator 2 left parenthesis a plus b right parenthesis left parenthesis b c minus a c right parenthesis plus left parenthesis a minus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
Again taking similar terms common
=> $fraction numerator left parenthesis a minus b right parenthesis left parenthesis 2 left parenthesis negative c right parenthesis left parenthesis a plus b right parenthesis plus left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $fraction numerator left parenthesis a minus b right parenthesis left parenthesis negative 2 a c minus 2 b c plus b c plus a b plus c squared plus a c right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $fraction numerator left parenthesis a minus b right parenthesis left parenthesis negative a c minus b c plus a b plus c squared right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
Again taking similar terms common
=> $fraction numerator left parenthesis a minus b right parenthesis left parenthesis a b minus b c plus c squared minus a c right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $fraction numerator left parenthesis a minus b right parenthesis left parenthesis left parenthesis c left parenthesis c minus b right parenthesis minus a left parenthesis c minus b right parenthesis right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $fraction numerator left parenthesis a minus b right parenthesis left parenthesis c minus a right parenthesis left parenthesis a minus c right parenthesis over denominator left parenthesis a plus b right parenthesis left parenthesis b plus c right parenthesis left parenthesis c plus a right parenthesis end fraction$
=> $negative x y z$

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