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Question

In $triangle A B C comma left parenthesis a minus b right parenthesis squared c o s squared invisible function application C divided by 2 plus left parenthesis a plus b right parenthesis squared s i n squared invisible function application C divided by 2 equals$

c
$c / 2$
2c
$c^{2}$

hint Hint:

In this question first we will use trigonometric formula $cos squared open parentheses C over 2 close parentheses equals 1 minus sin squared open parentheses C over 2 close parentheses$ then we will use the formula from solution of triangle $sin open parentheses C over 2 close parentheses equals square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis over denominator a b end fraction end root$ where $s equals fraction numerator a plus b plus c over denominator 2 end fraction$ .

The correct answer is: $c squared$

In this question we have to find the value of $left parenthesis a minus b right parenthesis squared cos to the power of 2 end exponent open parentheses fraction numerator C over denominator 2 end fraction close parentheses plus left parenthesis a plus b right parenthesis squared sin to the power of 2 end exponent open parentheses fraction numerator C over denominator 2 end fraction close parentheses$
$left parenthesis a squared plus b squared minus 2 a b right parenthesis cos squared open parentheses C over 2 close parentheses plus left parenthesis a squared plus b squared plus 2 a b right parenthesis sin squared open parentheses C over 2 close parentheses$
Step1: Using trigonometric formula.
$cos squared open parentheses C over 2 close parentheses equals 1 minus sin squared open parentheses C over 2 close parentheses$
$left parenthesis a squared plus b squared minus 2 a b right parenthesis left parenthesis 1 minus sin squared open parentheses C over 2 close parentheses right parenthesis plus left parenthesis a squared plus b squared plus 2 a b right parenthesis sin squared open parentheses C over 2 close parentheses$
$a squared plus b squared plus 2 a b left parenthesis 2 sin squared open parentheses C over 2 close parentheses minus 1 right parenthesis$
Step2: Using formula from Solution of Triangle.
$sin open parentheses C over 2 close parentheses equals square root of fraction numerator left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis over denominator a b end fraction end root$ where $s equals fraction numerator a plus b plus c over denominator 2 end fraction$
=> $a squared plus b squared plus 2 a b left parenthesis fraction numerator 2 left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis over denominator a b end fraction minus 1 right parenthesis$
by putting $a squared plus b squared plus 2 a b left parenthesis fraction numerator 2 left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis over denominator a b end fraction minus 1 right parenthesis$
=> $a squared plus b squared plus 2 a b left parenthesis fraction numerator left parenthesis c plus b minus a right parenthesis left parenthesis c plus a minus b right parenthesis over denominator 2 a b end fraction minus 1 right parenthesis$
=> $a squared plus b squared plus c squared minus left parenthesis b minus a right parenthesis squared minus 2 a b$
=> $c squared$
Hence, the value of the expression is $c squared$ .

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