A) \[\cot C\]
B) \[c\cot C\]
C) \[\cot \frac{C}{2}\]
D) \[c\cot \frac{C}{2}\]
Correct Answer: D
Solution :
\[\left\{ \cot \frac{A}{2}+\cot \frac{B}{2} \right\}\left\{ a{{\sin }^{2}}\frac{B}{2}+b{{\sin }^{2}}\frac{A}{2} \right\}\] = \[\left\{ \frac{\cos \frac{C}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}} \right\}\text{ }\left\{ a{{\sin }^{2}}\frac{B}{2}+b{{\sin }^{2}}\frac{A}{2} \right\}\] = \[\left\{ \cos \frac{C}{2} \right\}\text{ }\left\{ a\frac{\sin \frac{B}{2}}{\sin \frac{A}{2}}+b\frac{\sin \frac{A}{2}}{\sin \frac{B}{2}} \right\}\] \[=\sqrt{\frac{s(s-c)}{ab}}\left\{ a\frac{\sqrt{\frac{(s-a)(s-c)}{ac}}}{\sqrt{\frac{(s-b)(s-c)}{bc}}}+b\frac{\sqrt{\frac{(s-b)(s-c)}{bc}}}{\sqrt{\frac{(s-a)(s-c)}{ac}}} \right\}\] \[=\sqrt{\frac{s(s-c)}{ab}}\left\{ \sqrt{\left( \frac{s-a}{s-b} \right)\text{ }ab}+\sqrt{\left( \frac{s-b}{s-a} \right)\text{ }ab} \right\}\] =\[\sqrt{s(s-c)}\left\{ \frac{s-a+s-b}{\sqrt{(s-a)(s-b)}} \right\}=\sqrt{s(s-c)}\left\{ \frac{2s-a-b}{\sqrt{(s-a)(s-b)}} \right\}\] \[=c\sqrt{\frac{s(s-c)}{(s-a)(s-b)}}=c\cot \frac{C}{2}\]. Trick: Such type of unconditional problems can be checked by putting the particular values for \[a=1,\] \[b=\sqrt{3},\] \[c=2\] and \[A={{30}^{o}}\], \[B={{60}^{o}},C={{90}^{o}}\]. Hence expression is equal to 2 which is given by (d).
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