 Solution

The Pauli matrix multiplication law givesFor a unit vector $\hat{\mathbf{n}}$, it follows that $(\hat{\mathbf{n}}\cdot \mathbf{\sigma }{)}^2=I$. In the exponential series the even powers therefore multiply $I$ and the odd powers multiply $\hat{\mathbf{n}}\cdot \mathbf{\sigma }$. Summing them separately,Equivalently, the eigenvalues of $\hat{\mathbf{n}}\cdot \mathbf{\sigma }$ are $\pm 1$, so its two spectral projectors acquire phases $e^{\mp i\theta /2}$. The half-angle belongs to a spinor rotation, while the associated physical spin vector rotates through $\theta$.