Proof:

If $G$ is connected then so is $G/K$ since the quotient map $\eta: G \longrightarrow G/K$ is a continuous surjection. To prove the converse suppose that $K$ and $G/K$ are connected and $f : G \longrightarrow \{0, 1\}$ be an arbitrary continuous map. We have to show that $f$ is constant. The restriction of $f$ to $K$ must be constant and since each coset $gK$ is connected, $f$ must be constant on $gK$ as well taking value $f(g)$. Thus we have a well defined map $\tilde f: G/K \longrightarrow \{0, 1\}$ such that $\tilde{f}\circ \eta = f$. By the fundamental property of quotient spaces, it follows that $\tilde f$ is continuous and so must be constant since $G/K$ is connected. Hence $f$ is also constant and we conclude that $G$ is connected. $\square$

Since we shall not need (2), we shall omit the proof. A proof is available in [12], p. 109.