Question:
Prove that $1+i^{2}+i^{4}+i^{6}=0$
Solution:
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.
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