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Question

In straight triangle A B C comma left parenthesis a plus b plus c right parenthesis open parentheses tan A over 2 plus tan space B over 2 close parentheses equals

  1. 2ccotA over 2
  2. 2ccotB over 2
  3. 2ccotC over 2
  4. 2ctanC over 2

hintHint:

The angles of the triangle ABC are denoted by A, B, C and the corresponding opposite sides by a, b, c. s denotes the semi-perimeter of the triangle ABC, ∆ its area and R the radius of the circle circumscribing the triangle ABC i.e., R is the circum-radius.

The correct answer is: 2ccotC over 2


    straight triangle A B C comma left parenthesis a plus b plus c right parenthesis open parentheses tan A over 2 plus tan space B over 2 close parentheses
W e space k n o w space t h a t space semicolon space tan A over 2 equals square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root space semicolon space space tan B over 2 equals square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root space semicolon 2 s equals left parenthesis a plus b plus c right parenthesis
left parenthesis a plus b plus c right parenthesis space tan A over 2 plus space tan B over 2 equals left parenthesis square root of fraction numerator left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus a right parenthesis end fraction end root plus square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus c right parenthesis over denominator s left parenthesis s minus b right parenthesis end fraction end root right parenthesis left parenthesis 2 s right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 2 s square root of fraction numerator left parenthesis s minus c right parenthesis over denominator s end fraction end root left parenthesis space fraction numerator s minus a plus s minus b over denominator square root of left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis end root end fraction space right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals open parentheses 2 c close parentheses 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 space space space space space space space space space space space space space space space space space space space space space space left curly bracket bold space square root of fraction numerator bold s bold left parenthesis bold s bold minus bold c bold right parenthesis over denominator bold left parenthesis bold s bold minus bold a bold right parenthesis bold left parenthesis bold s bold minus bold b bold right parenthesis end fraction end root bold space bold equals bold space bold italic c bold italic o bold italic t bold C over bold 2 space right curly bracket
space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses 2 c close parentheses c o t C over 2
     
     
     
     
     

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