$\def\sign{\operatorname{sign}}$
For homework, I am trying to show that the map $\sign:S_n \to \{\pm 1\}$ is multiplicative, i.e. that for any permutations $\sigma_1,\sigma_2$ in the symmetric group $S_n$, we have $\sign(\sigma_1 \sigma_2) = \sign\sigma_1 \sign\sigma_2.$
The definition for $\sign$ that I am using is that if $\sigma = \gamma_1\cdots \gamma_k$ is the cycle decomposition of $\sigma \in S_n$ and $\ell_1,\ldots,\ell_k$ are the cycle lengths of $\gamma_1,\ldots,\gamma_k$ respectively, then
$\sign(\sigma):= (-1)^{\ell_1-1}\cdots(-1)^{\ell_k-1}.$
I showed first that the formula holds for two transpositions. Then I showed that it holds for a transposition and a cycle.
However, I got stuck trying to show that the formula holds for a transposition and a product of two cycles, i.e. $\sigma_1 = \tau$ and $\sigma_2 = \gamma_1\gamma_2$. I feel like this case is much more complicated than the others which makes me think I am taking the wrong approach.
If $\tau,\gamma_1,\gamma_2$ are all disjoint then the formula holds trivially because then $\gamma_1\gamma_2\tau$ is the cycle decomposition of $\sigma_1\sigma_2$. Otherwise they are not all disjoint, at which point it seems to get complicated very quickly and I don't know how to proceed.
Would someone please help me understand the best approach to this proof?