Let $c = (1, 2, \dotsc, n)$. We see that
\begin{align*}
c (1, 2) c^{-1} &= (2, 3) \\
c (2, 3) c^{-1} &= (3, 4) \\
&\vdots \\
c (n-2, n-1) c^{-1} &= (n-1, n),
\end{align*}
so that $(i, i+1) \in \langle (1, 2), c \rangle$ for all $1 \leq i \leq n-1$. Next, we have
\begin{align*}
(2, 3) (1, 2) (2, 3)^{-1} &= (1, 3) \\
(3, 4) (1, 3) (3, 4)^{-1} &= (1, 4) \\
&\vdots \\
(n-1, n) (1, n-1) (n-1, n)^{-1} &= (1, n),
\end{align*}
so that $(1, i) \in \langle (1, 2), c \rangle$ for all $1 \leq i \leq n$. Choose any $1 \leq i < j \leq n$, then
$$ (i, j) = (1, i) (1, j) (1, i)^{-1} \in \langle (1, 2), c \rangle. $$
Therefore, $\langle (1, 2), c \rangle$ contains all transpositions. Hence, $\langle (1, 2), c \rangle = S_n$.