Exponential of Pauli spin matrices

[Happiness]

How do we get (6.265)?

Shouldn't we have

$exp(-i\frac{\alpha}{2}\hat{n}.\sigma)=\cos(\frac{\alpha}{2}\hat{n}.\sigma)-i\sin(\frac{\alpha}{2}\hat{n}.\sigma)$?

 

[blue_leaf77]

By (6.241), you have $(\mathbf{\sigma} \cdot \hat{n} )^2 = 1$. Use this identity in the Taylor expansions of $\cos(\frac{\alpha}{2}\hat{n}.\sigma)$ and $\sin(\frac{\alpha}{2}\hat{n}.\sigma)$ to reduce the higher powers.