Provide detailed proofs of the trace identities:

(a) traces of products of \(\gamma\) matrices (with no \(\gamma^{\mu \dagger}\) ) are representation independent (same for \(\gamma^{\mu}\) and \(\left.\bar{\gamma}^{\mu} \equiv S \gamma^{\mu} S^{-1}\right)\).

(b) \(\operatorname{tr}(I)=4, \operatorname{tr}\left(\gamma^{\mu} \gamma^{u}\right)=4 g^{\mu u}, \operatorname{tr}\left(\gamma^{\mu}\right)=\operatorname{tr}\left(\right.\) odd \(\#\) of \(\left.\gamma^{\mu \prime} \mathrm{s}\right)=\operatorname{tr}\left(\sigma^{\mu u}\right)=0\), \(\operatorname{tr}\left(\gamma^{\mu} \gamma^{u} \gamma^{ho} \gamma^{\sigma}\right)=4\left(g^{\mu u} g^{ho \sigma}-g^{\mu ho} g^{u \sigma}+g^{\mu \sigma} g^{u ho}\right)\).

(c) \(\operatorname{tr}\left(\gamma^{5} \gamma^{\mu} \gamma^{u}\right)=\operatorname{tr}\left(\gamma^{5}\right.\) (odd \# of \(\gamma^{\mu \prime}\) s \(\left.)\right)=0\) and \(\operatorname{tr}\left(\gamma^{5} \gamma^{\mu} \gamma^{u} \gamma^{ho} \gamma^{\sigma}\right)=-4 i \epsilon^{\mu u ho \sigma}\).