If $Z_1,Z_2$  are two complex numbers then $|Z_1+\sqrt{Z_1^2-Z_2^2}|+|Z_1-\sqrt{Z_1^2-Z_2^2}|=\mathrm{......}$ 

Let $\alpha =Z_1+\sqrt{Z_1^2+Z_2^2}$ 

$\beta =Z_1-\sqrt{Z_1^2-Z_2^2}$

$\alpha +\beta =2Z_1$

$\alpha -\beta =2\sqrt{Z_1^2-Z_2^2}$

$\alpha \beta ={\left(Z_2\right)}^2$

Let $E={(|Z_1+Z_2|+|Z_1-Z_2|)}^2$

 $E=2{\left|Z_1\right|}^2+2{\left|Z_2\right|}^2$ $+2\left|Z_1^2-Z_2^2\right|$

$E={\left|\frac{\alpha +\beta }{2}\right|}^2+{\left|\frac{\alpha -\beta }{2}\right|}^2$  + $2\left|\alpha \beta \right|$

$={\left(\left|\alpha \right|+\left|\beta \right|\right)}^2$