Show $\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A,A\in M(2,\mathbb{C})$ 
Show $$\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A$$
  for $A\in M(2,\mathbb{C})$. In addition, $\operatorname{trace}(A)=0$.

Can anyone give me a hint how this can connect with cosine and sine? Thanks!

By Cayley-Hamilton theorem, $A^2$ is a constant or degree-$1$ polynomial in $A$. It follows that $\exp(A)=pI+qA$ for some $p,q\in\mathbb{C}$. Since $A$ is traceless, its two eigenvalues are $iz$ and $-iz$ for some $z\in\mathbb{C}$ with $\det(A)=z^2$. Without loss of generality, we take $\color{red}{\sqrt{\det(A)}=z}$. Take the traces of both sides of $\exp(A)=pI+qA$, we get $e^{iz}+e^{-iz}=2p$, i.e. $p=\frac12\left(e^{iz}+e^{-iz}\right)=\cos z$. To find $q$, consider an eigenvector of $A$ corresponding to the eigenvalue $iz$. We have $e^{iz}v=\exp(A)v=(pI+qA)v=\cos(z)v+iqzv$. Since $e^{iz}=\cos z + i\sin z$, we get $q=\frac{\sin z}{z}$.
