prove that

L.H.S. $=1+i^{2}+i^{4}+i^{6}$

To Prove: $1+i^{2}+i^{4}+i^{6}=0$

$\Rightarrow 1+(-1)+1+i^{2}$

Since, $i^{4 n}=1$

(Where n is any positive integer)

$\Rightarrow i^{4 n+2}$

$\Rightarrow i^{2}=-1$

$\Rightarrow 1+-1+1+-1=0$

Hence proved.