Maths-
General
Easy

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 =

  1. cos squared space A
  2. cos squared space B
  3. sin squared space A
  4. sin squared space B

Hint:

In this question, we have to find 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. For this we will use some trigonometric formula s equals fraction numerator a plus b plus c over denominator 2 end fraction comma space sin space 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 b c end fraction end root space comma space cos space A over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator b c end fraction end root space a n d space sin space 2 A space equals 2. sin space A. cos space A to find the required value.

The correct answer is: sin squared space A


    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 space space space space space space space space space space
equals fraction numerator 2 s space left parenthesis 2 s minus a minus a right parenthesis left parenthesis 2 s minus b minus b right parenthesis left parenthesis 2 s minus c minus c right parenthesis over denominator 4 b squared c squared end fraction space space space space space space space space space space space space space space space open square brackets s equals fraction numerator a plus b plus c over denominator 2 end fraction close square brackets
equals fraction numerator 2 s.2 left parenthesis s minus a right parenthesis.2 left parenthesis s minus b right parenthesis.2 left parenthesis s minus c right parenthesis over denominator 4 b squared c squared end fraction
equals fraction numerator 4 s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator b squared c squared end fraction
equals 4 sin squared A over 2 cos squared A over 2 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 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 open square brackets sin space 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 b c end fraction end root comma space cos A over 2 equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator b c end fraction end root close square brackets
equals sin squared A 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 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 space space space space space space space space space space space space space space space space open square brackets sin space 2 A equals 2. sin space A. cos space B close square brackets

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