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Question

In $triangle A B C comma sum left parenthesis a squared s i n invisible function application left parenthesis B minus C right parenthesis right parenthesis divided by left parenthesis s i n invisible function application A right parenthesis equals$

$a^{2} b^{2}$
$b^{2} c^{2}$
$c^{2} a^{2}$
0

hint Hint:

In this question first we will us Sine Rule. Then we use the trigonometric formula $sin open parentheses A plus B close parentheses sin open parentheses A minus B close parentheses equals sin squared open parentheses A close parentheses minus sin squared open parentheses B close parentheses$ to find the value of the sum

The correct answer is: 0

In this question we have to find the value of $fraction numerator a squared sin open parentheses B minus C close parentheses over denominator sin open parentheses A close parentheses end fraction plus fraction numerator b squared sin open parentheses C minus A close parentheses over denominator sin open parentheses B close parentheses end fraction plus fraction numerator c squared sin open parentheses A minus B close parentheses over denominator sin open parentheses C close parentheses end fraction$
Step1: Using Sine Rule
$fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals fraction numerator b over denominator sin open parentheses B close parentheses end fraction equals fraction numerator c over denominator sin open parentheses C close parentheses end fraction equals k$
=> $a equals space k sin open parentheses A close parentheses comma b equals k sin open parentheses B close parentheses comma space c equals k sin open parentheses C close parentheses$
By substituting these values in the given expression we get.
$fraction numerator k squared sin squared open parentheses A close parentheses sin open parentheses B minus C close parentheses over denominator sin open parentheses A close parentheses end fraction plus fraction numerator k squared sin squared open parentheses B close parentheses sin open parentheses C minus A close parentheses over denominator sin open parentheses B close parentheses end fraction plus fraction numerator k squared sin squared open parentheses C close parentheses sin open parentheses A minus B close parentheses over denominator sin open parentheses C close parentheses end fraction$
=> $k squared left parenthesis sin open parentheses A close parentheses sin open parentheses B minus C close parentheses plus sin open parentheses B close parentheses sin open parentheses C minus A close parentheses plus sin open parentheses C close parentheses sin open parentheses A minus B close parentheses right parenthesis$
we know that $A plus B plus C equals straight pi$
=> $A equals straight pi minus left parenthesis straight B plus straight C right parenthesis$
=> $sin open parentheses A close parentheses equals sin open parentheses B plus C close parentheses$
So, we can write $sin open parentheses A close parentheses equals sin open parentheses B plus C close parentheses comma sin open parentheses B close parentheses equals sin open parentheses A plus C close parentheses comma sin open parentheses C close parentheses equals sin open parentheses A plus B close parentheses$
by substituting these values in the above expression we get,
=> $k squared left parenthesis sin open parentheses B plus C close parentheses sin open parentheses B minus C close parentheses plus sin open parentheses C plus A close parentheses sin open parentheses C minus A close parentheses plus sin open parentheses A plus B close parentheses sin open parentheses A minus B close parentheses right parenthesis$
Step2: Using trigonometric formula
We know that $sin open parentheses A plus B close parentheses sin open parentheses A minus B close parentheses equals sin squared open parentheses A close parentheses minus sin squared open parentheses B close parentheses$
=> $k squared left parenthesis sin squared open parentheses B close parentheses minus sin squared open parentheses C close parentheses plus sin squared open parentheses C close parentheses minus sin squared open parentheses A close parentheses plus sin squared open parentheses A close parentheses minus sin squared open parentheses B close parentheses right parenthesis$
=> $0$
Hence, the value of sum is $0$ .

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