Maths-
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Question

In any straight triangle A B C fraction numerator left parenthesis a plus b plus c right parenthesis left parenthesis b plus c minus a right parenthesis left parenthesis c plus a minus b right parenthesis left parenthesis a plus b minus c right parenthesis over denominator 4 b squared c squared end fraction equals

  1. cos squared space A
  2. cos squared space B
  3. S i n squared space A
  4. S i n squared space B

hintHint:

We will discuss the properties of triangle formulae which will help us to solve In space straight triangle A B C comma space fraction numerator left parenthesis a plus b plus c right parenthesis left parenthesis b plus c minus a right parenthesis left parenthesis c plus a minus b right parenthesis left parenthesis a plus b minus c right parenthesis over denominator 4 b squared c squared end fraction of problems on triangle.

The correct answer is: S i n squared space A


    In any straight triangle A B C fraction numerator left parenthesis a plus b plus c right parenthesis left parenthesis b plus c minus a right parenthesis left parenthesis c plus a minus b right parenthesis left parenthesis a plus b minus c right parenthesis over denominator 4 b squared c squared end fraction equals
    S =fraction numerator a plus b plus c over denominator 2 end fraction { semi-perimeter }
    left curly bracket left parenthesis a plus b plus c equals 2 s comma left parenthesis b plus c minus a equals 2 s minus 2 a space comma c plus a minus b equals 2 s minus 2 b space comma a plus b minus c equals 2 s minus 2 c space right curly bracket
fraction numerator left parenthesis 2 s right parenthesis left parenthesis 2 s minus 2 a right parenthesis left parenthesis 2 s minus 2 b right parenthesis left parenthesis 2 s minus 2 c right parenthesis over denominator 4 b squared c squared end fraction equals space fraction numerator 4 left parenthesis s right parenthesis left parenthesis s minus a right parenthesis cross times left parenthesis s minus 2 b right parenthesis left parenthesis s minus 2 c right parenthesis over denominator open parentheses b c close parentheses open parentheses b c close parentheses end fraction
bold space bold left square bracket bold space bold space bold italic W bold italic e bold space bold italic k bold italic n bold italic o bold italic w bold space bold italic t bold italic h bold italic a bold italic t bold space bold italic s bold italic i bold italic n bold space bold A over bold 2 bold space bold equals bold space square root of fraction numerator bold left parenthesis bold s bold minus bold b bold right parenthesis bold left parenthesis bold s bold minus bold c bold right parenthesis over denominator bold b bold c end fraction end root bold comma bold space bold space bold italic c bold italic o bold italic s bold space bold A over bold 2 bold space bold equals bold space square root of fraction numerator bold s bold left parenthesis bold s bold minus bold a bold right parenthesis over denominator bold b bold c end fraction end root bold space bold right square bracket
fraction numerator 4 left parenthesis s right parenthesis left parenthesis s minus a right parenthesis cross times left parenthesis s minus 2 b right parenthesis left parenthesis s minus 2 c right parenthesis over denominator open parentheses b c close parentheses open parentheses b c close parentheses end fraction bold space equals bold space 4 cross times s i n squared space straight A over 2 space cross times c o s squared space straight A over 2 space
left parenthesis bold space 2 cross times s i n space straight A over 2 space cross times c o s space straight A over 2 space right parenthesis squared space space space space space space space space space space left curly bracket W e space k n o w space t h a t space colon space 2 cross times s i n space straight A over 2 space cross times c o s space straight A over 2 equals space sin space A space right curly bracket
sin squared A

    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.

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