Sorry the symbols in the first thread didnt work. Here is the first question again.
Thus, $\displaystyle \sigma = (1,2,3,4)$ and $\displaystyle \tau = (2,4)$. A simple calculation will show $\displaystyle \tau \sigma = \sigma^3 \tau$. But $\displaystyle \sigma^3 = \sigma^{-1}$. Thus, $\displaystyle \tau \sigma = \sigma^{-1} \tau$. This means, $\displaystyle \tau \sigma^{k} = \tau \sigma \sigma^{k-1} = \sigma^{-1} \tau \sigma^{k-1} = \sigma^{-1} \tau \sigma \sigma^{k-2} = \sigma^{-2} \tau \sigma^{k-2} = ... = \sigma^{-k} \tau$.