= Solution
Besides their squares, direct multiplication yields $\sigma_x\sigma_y=i\sigma_z$, $\sigma_y\sigma_z=i\sigma_x$ and $\sigma_z\sigma_x=i\sigma_y$. Reversing each pair changes the sign. These identities combine into the <Pauli matrix multiplication identity>
$$
\sigma_i\sigma_j=\delta_{ij}I+i\sum_k\epsilon_{ijk}\sigma_k.
$$
Adding the expression with $i,j$ exchanged cancels the antisymmetric term, proving $\{\sigma_i,\sigma_j\}=2\delta_{ij}I$. Taking the <trace> instead uses $\operatorname{tr}I=2$ and $\operatorname{tr}\sigma_k=0$, proving $\operatorname{tr}(\sigma_i\sigma_j)=2\delta_{ij}$.
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