In a $△\mathrm{ABC},\mathrm{ }{\left(\mathrm{a}-\mathrm{b}\right)}^2{\cos }^2 \frac{\mathrm{C}}{2}+{\left(\mathrm{a}+\mathrm{b}\right)}^2{\sin }^2 \frac{\mathrm{C}}{2}$

The given equation is ${\left(a-b\right)}^2{\cos }^2\left(\frac{C}{2}\right)+{\left(a+b\right)}^2{\sin }^2\left(\frac{C}{2}\right)$

Simplifying 

$\begin{array}{rcl} & & \left(a^2+b^2-2ab\right){\cos }^2\left(\frac{C}{2}\right)+\left(a^2+b^2+2ab\right){\sin }^2\left(\frac{C}{2}\right)\\ & =& \left(a^2+b^2\right)-2ab\cos C\\ & =& c^2\end{array}$