a) Prove that $\text{Tr}(\gamma^5 \not a \not b) = 0,$ where $\not a = a_{\nu}\gamma^{\nu}$.
b) Prove that $\text{Tr}(\gamma^5 \not a \not b \not c \not d) = 4i\epsilon_{\alpha \beta \gamma \delta} a^{\alpha}b^{\beta}c^{\gamma}d^{\delta}$
For (a), I have tried writing $$\text{Tr}(\gamma^5 \gamma^{\mu} \gamma^{\nu}) = \frac{1}{2} \text{Tr}(\left\{\gamma^0, \gamma^1 \gamma^2 \gamma^3 \gamma^{\mu}\gamma^{\nu}\right\}),$$ making use of the explicit form for $\gamma^5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3$. I played around with shifting the ordering of the terms there (and therefore introducing minus signs) to try and get a result that gave me that $\text{Tr} = - \text{Tr}$ and thus infer it was zero but I have not managed.
Any hints on how to progress?
Thanks!