Is $e^{\sin(\sin (z))}$ analytic?

Question

I have tried searching the internet for any possible solutions to try and find an answer to this question. One idea that I had was to use Hyperbolic functions to try and get the form $z = u + iy$. But the question is:

$$e^{\sin(\sin (z))}$$

I couldn't get past the 'e' part. Not only that, I am having trouble with the hyperbolic substitutions and simplifying them in a way that results in $z = u + iy$. Is there another way to show whether this expression is analytic or not? Or is it possible to prove it through $U_x=V_y$ and $U_y= - V_x$ ?

Do forgive me if it my question is a bit confusing.

Answer 1

Hint: You can use this version of Cauchy-Riemann instead:

$$\frac{\partial f}{\partial x} = -i\frac{\partial f}{\partial y}$$

Answer 2

Both $e^z$ and $\sin z$ are entire (i.e. analytic everywhere) on $\Bbb C$, so is any order of their combination.