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!

A possible method, if you so choose to accept it, is to do the following: Define the matrix valued functions 
$$
u(t) = \exp(tA)
$$
and
$$
v(t) = \cos\left(\sqrt{\det(tA)}\right)I + \frac{\sin\left(\sqrt{\det(tA)}\right)}{\sqrt{\det(tA)}}tA
$$
Then show that $u(0)=v(0)$ and $u'(t) = v'(t)$. This will imply that $u(t)=v(t)$ for all $t \geq 0$, in particular for $t=1$.
Along the way it might help to remember that $\sqrt{\det(tA)} = t \sqrt{\det(A)}$ (since $A$ is $2 \times 2$). 
