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Maths-

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Question

$I n space a n y space triangle A B C left parenthesis left parenthesis a plus b plus c right parenthesis left parenthesis b plus c minus a right parenthesis left parenthesis c plus a minus b right parenthesis left parenthesis a plus b minus c right parenthesis right parenthesis divided by left parenthesis 4 b squared c squared right parenthesis$

$c o s^{2} A$
$c o s^{2} B$
$s i n^{2} A$
$s i n^{2} B$

hint Hint:

In this question first we have to rewrite the expression in terms of semi perimeter s and the relation is as follows $s equals fraction numerator a plus b plus c over denominator 2 end fraction$ then we will use the formula $cos squared open parentheses A over 2 close parentheses sin squared open parentheses A over 2 close parentheses equals fraction numerator s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator b c cross times b c end fraction$ to find the value of the expression.

The correct answer is: $s i n squared invisible function application A$

Step1: Arranging the expression in terms of perimeter.
We know that $s equals fraction numerator a plus b plus c over denominator 2 end fraction$ where s is the semi-perimeter of the triangle.
We can also write $a plus b equals space 2 s minus c b plus c equals 2 s minus a a plus c equals 2 s minus b$
Step2: Rewriting the expression
$fraction numerator left parenthesis 2 s right parenthesis left parenthesis 2 s minus a minus a right parenthesis left parenthesis 2 s minus b minus b right parenthesis left parenthesis 2 s minus c minus c right parenthesis over denominator 4 cross times b c cross times b c end fraction$
=> $fraction numerator 4 s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator b c cross times b c end fraction$
We know that the formula $cos squared open parentheses A over 2 close parentheses sin squared open parentheses A over 2 close parentheses equals fraction numerator s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator b c cross times b c end fraction$
=> $4 cos squared open parentheses A over 2 close parentheses sin squared open parentheses A over 2 close parentheses$
We know that $sin open parentheses theta close parentheses space equals space 2 sin open parentheses theta over 2 close parentheses cos open parentheses theta over 2 close parentheses$
=> $sin squared open parentheses A close parentheses$
Hence, the value of the expression is $sin squared open parentheses A close parentheses$ .

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