I have proved that unit sphere is connected in R^3, can I use this fact to prove that complement of unit sphere is not connected?
No. Instead, try assuming for contradiction that there is a path $\gamma\colon I\to\mathbb R^3$ going from, say, $\mathbf 0$ to some $x\in\mathbb R^3$ with $|x|>1$. Then use a connectedness argument to show that the continuous map $|\gamma |$ (composite of $\gamma$ and the norm $|\cdot |$) must pass through $1$. This is sufficient.