In a $△ABC,\left(a+b+c\right)\left(\tan \frac{B}{2}+\tan \frac{C}{2}\right)=$

The given expression is $\left(a+b+c\right)\left(\tan \left(\frac{B}{2}\right)+\tan \left(\frac{C}{2}\right)\right)$

Use the formulae from properties of triangle to simplify the above expression

$\begin{array}{rcl}\left(a+b+c\right)\left(\tan \left(\frac{B}{2}\right)+\tan \left(\frac{C}{2}\right)\right)& =& 2s\left(\frac{∆}{s\left(s-b\right)}+\frac{∆}{s\left(s-c\right)}\right)\\ & =& \frac{2∆s}{s}\left(\frac{s-b+s-c}{\left(s-b\right)\left(s-c\right)}\right)\\ & =& 2∆\left(\frac{a}{\left(s-b\right)\left(s-c\right)}\right)\\ & =& 2\frac{∆}{bc}\left(\frac{abc}{\left(s-b\right)\left(s-c\right)}\right)\\ & =& 2acot\left(\frac{A}{2}\right)\end{array}$