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In a triangle $\mathrm{ABC}$, if $\mathrm{a}-2 \mathrm{~b}+\mathrm{c}=0$, then $\cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{C}{2}\right)=$
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$3$
We are given that $a-2 b+c=0 \Rightarrow 2 b=a+c$
$\Rightarrow \mathrm{b}=\frac{\mathrm{a}+\mathrm{c}}{2}$
Now $\cot \left(\frac{\mathrm{A}}{2}\right) \times \cot \left(\frac{\mathrm{c}}{2}\right)=\left[\frac{\cos \left(\frac{\mathrm{A}}{2}\right)}{\sin \left(\frac{\mathrm{A}}{2}\right)}\right] \cdot\left[\frac{\cos \left(\frac{\mathrm{c}}{2}\right)}{\sin \left(\frac{\mathrm{c}}{2}\right)}\right]$
$\Rightarrow \cot \left(\frac{\mathrm{A}}{2}\right) \times \cot \left(\frac{\mathrm{c}}{2}\right)=\left[\frac{2 \cos \left(\frac{\mathrm{A}}{2}\right) \times \sin \left(\frac{\mathrm{A}}{2}\right)}{2 \sin ^2\left(\frac{\mathrm{A}}{2}\right)}\right]$
$\left[\frac{2 \cos ^2\left(\frac{\mathrm{c}}{2}\right)}{2 \sin \left(\frac{\mathrm{c}}{2}\right) \times \mathrm{WS}\left(\frac{\mathrm{c}}{2}\right)}\right]$
$\begin{aligned} & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left[\frac{\sin A}{1-\cos A}\right] \times \frac{1+\cos c}{(\sin c)} \\ & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left(\frac{k \sin A}{k \sin C}\right) \times\left[\frac{1+\cos c}{1-\cos A}\right]\end{aligned}$
$\left\{\begin{array}{l}\text { By Sin ce rule } \\ \frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\operatorname{Sin} \mathrm{B}}=\frac{\mathrm{c}}{\operatorname{Sin} \mathrm{C}}=\mathrm{k}\end{array}\right\}$
$\Rightarrow \cot \left(\frac{\mathrm{A}}{2}\right) \cdot \cot \left(\frac{\mathrm{c}}{2}\right)=\left(\frac{\mathrm{a}}{\mathrm{c}}\right) \times \frac{1+\left[\frac{\mathrm{a}^2+\mathrm{b}^2-\mathrm{c}^2}{2 \mathrm{ab}}\right]}{1-\left[\frac{\mathrm{b}^2+\mathrm{c}^2-\mathrm{a}^2}{2 \mathrm{bc}}\right]}$
$\begin{aligned} & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left(\frac{a}{c}\right) \times\left[\frac{\left(2 a b+a^2+b^2-c^2\right) 2 b c}{2 a b\left(2 b c-b^2-c^2+a^2\right)}\right] \\ & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left[\frac{2 a b+a^2+b^2-c^2}{2 b c-b^2-c^2+a^2}\right]\end{aligned}$
$\Rightarrow \mathrm{b}=\frac{\mathrm{a}+\mathrm{c}}{2}$
Now $\cot \left(\frac{\mathrm{A}}{2}\right) \times \cot \left(\frac{\mathrm{c}}{2}\right)=\left[\frac{\cos \left(\frac{\mathrm{A}}{2}\right)}{\sin \left(\frac{\mathrm{A}}{2}\right)}\right] \cdot\left[\frac{\cos \left(\frac{\mathrm{c}}{2}\right)}{\sin \left(\frac{\mathrm{c}}{2}\right)}\right]$
$\Rightarrow \cot \left(\frac{\mathrm{A}}{2}\right) \times \cot \left(\frac{\mathrm{c}}{2}\right)=\left[\frac{2 \cos \left(\frac{\mathrm{A}}{2}\right) \times \sin \left(\frac{\mathrm{A}}{2}\right)}{2 \sin ^2\left(\frac{\mathrm{A}}{2}\right)}\right]$
$\left[\frac{2 \cos ^2\left(\frac{\mathrm{c}}{2}\right)}{2 \sin \left(\frac{\mathrm{c}}{2}\right) \times \mathrm{WS}\left(\frac{\mathrm{c}}{2}\right)}\right]$
$\begin{aligned} & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left[\frac{\sin A}{1-\cos A}\right] \times \frac{1+\cos c}{(\sin c)} \\ & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left(\frac{k \sin A}{k \sin C}\right) \times\left[\frac{1+\cos c}{1-\cos A}\right]\end{aligned}$
$\left\{\begin{array}{l}\text { By Sin ce rule } \\ \frac{\mathrm{a}}{\sin \mathrm{A}}=\frac{\mathrm{b}}{\operatorname{Sin} \mathrm{B}}=\frac{\mathrm{c}}{\operatorname{Sin} \mathrm{C}}=\mathrm{k}\end{array}\right\}$
$\Rightarrow \cot \left(\frac{\mathrm{A}}{2}\right) \cdot \cot \left(\frac{\mathrm{c}}{2}\right)=\left(\frac{\mathrm{a}}{\mathrm{c}}\right) \times \frac{1+\left[\frac{\mathrm{a}^2+\mathrm{b}^2-\mathrm{c}^2}{2 \mathrm{ab}}\right]}{1-\left[\frac{\mathrm{b}^2+\mathrm{c}^2-\mathrm{a}^2}{2 \mathrm{bc}}\right]}$
$\begin{aligned} & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left(\frac{a}{c}\right) \times\left[\frac{\left(2 a b+a^2+b^2-c^2\right) 2 b c}{2 a b\left(2 b c-b^2-c^2+a^2\right)}\right] \\ & \Rightarrow \cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{c}{2}\right)=\left[\frac{2 a b+a^2+b^2-c^2}{2 b c-b^2-c^2+a^2}\right]\end{aligned}$
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