[UPSC CSE 2018] Angular momentum, spin etc (Q285, 7+8=15M)

Prove the following identities:

(i) [\hat{p}_x,\hat{L}_y]=i\hbar\hat{p}_z

(ii) e^{i(\hat{\sigma}\cdot\hat{n})\theta}=\cos\theta+i(\hat{\sigma}\cdot\hat{n})\sin\theta