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Question

In $triangle A B C comma left parenthesis a left parenthesis cos invisible function application B plus cos invisible function application C right parenthesis right parenthesis divided by left parenthesis 2 left parenthesis b plus c right parenthesis right parenthesis equals$

$c o s^{2} A / 2$
$s i m^{2} A / 2$
$t a n^{2} A / 2$
$c o t^{2} A / 2$

hint Hint:

In this question first we have to use Sin Rule $a space equals space k sin open parentheses A close parentheses comma space b equals k sin open parentheses B close parentheses comma space c equals k sin open parentheses C close parentheses$ . Then we will use the C-D formula and Half- Angle formula
$sin open parentheses C close parentheses plus sin open parentheses D close parentheses equals 2 sin open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses cos open parentheses fraction numerator C minus D over denominator 2 end fraction close parentheses cos open parentheses C close parentheses plus cos open parentheses D close parentheses equals 2 cos open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses cos open parentheses fraction numerator C minus D over denominator 2 end fraction close parentheses$
$sin open parentheses 2 x close parentheses equals 2 sin open parentheses x close parentheses cos open parentheses x close parentheses$

The correct answer is: $s i m squared invisible function application A divided by 2$

In this question we have to find the value of $fraction numerator a left parenthesis cos open parentheses B close parentheses plus cos open parentheses C close parentheses right parenthesis over denominator 2 left parenthesis b plus c right parenthesis end fraction$
By using Sine Rule
$fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals fraction numerator b over denominator sin open parentheses B close parentheses end fraction equals fraction numerator c over denominator sin open parentheses C close parentheses end fraction equals k$
=> $a space equals space k sin open parentheses A close parentheses comma space b equals k sin open parentheses B close parentheses comma space c equals k sin open parentheses C close parentheses$
=> $fraction numerator k sin open parentheses A close parentheses left parenthesis cos open parentheses B close parentheses plus cos open parentheses C close parentheses right parenthesis over denominator 2 left parenthesis k sin open parentheses B close parentheses plus k sin open parentheses C close parentheses right parenthesis end fraction$
=> $fraction numerator sin open parentheses A close parentheses left parenthesis cos open parentheses B close parentheses plus cos open parentheses C close parentheses right parenthesis over denominator 2 left parenthesis sin open parentheses B close parentheses plus sin open parentheses C close parentheses right parenthesis end fraction$
By Using C-D formula and half-angle formula
$sin open parentheses C close parentheses plus sin open parentheses D close parentheses equals 2 sin open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses cos open parentheses fraction numerator C minus D over denominator 2 end fraction close parentheses cos open parentheses C close parentheses plus cos open parentheses D close parentheses equals 2 cos open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses cos open parentheses fraction numerator C minus D over denominator 2 end fraction close parentheses$
$sin open parentheses 2 x close parentheses equals 2 sin open parentheses x close parentheses cos open parentheses x close parentheses$
=> $fraction numerator 2 sin open parentheses begin display style A over 2 end style close parentheses cos open parentheses begin display style A over 2 end style close parentheses left parenthesis 2 cos open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses cos open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses right parenthesis over denominator 2 left parenthesis 2 sin open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses cos open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses right parenthesis end fraction$
We know that
$A plus B plus C space equals space pi$
$B plus C equals space pi space minus space A$
$fraction numerator B plus C over denominator 2 end fraction equals pi over 2 minus A over 2$
=> $sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals sin open parentheses pi over 2 minus A over 2 close parentheses$
=> $sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals cos open parentheses A over 2 close parentheses$
Similarly $cos open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals sin open parentheses A over 2 close parentheses$
By substituting these values in the given expression, we get
=> $fraction numerator sin open parentheses begin display style A over 2 end style close parentheses cos open parentheses begin display style A over 2 end style close parentheses sin open parentheses begin display style A over 2 end style close parentheses over denominator cos open parentheses begin display style A over 2 end style close parentheses end fraction$
=> $sin squared open parentheses A over 2 close parentheses$

Related Questions to study

In $triangle A B C comma left parenthesis left parenthesis b minus c right parenthesis divided by a right parenthesis sin invisible function application left parenthesis left parenthesis B plus C right parenthesis divided by 2 right parenthesis equals$

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parallel

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parallel

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

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Chemistry-General

parallel

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parallel

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parallel

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