I am having a lot of difficulty understanding what is meant to be holomorphic and how to even show. For example, I was asked to show that $$f(z)=\sin(z)-\frac{z^2}{z+1}$$ is holomorphic wherever it is defined. I will show you what I did try and where my problems are.
I tried to break it into pieces and x and y I used that $$\sin(z)=\sin(x)\cosh(y)+i\sinh(y)\cos(x)$$
and $$\frac{z^2}{z+1}=\frac{x^2-y^2+x^3+xy^2}{(1+x)^2+y^2}+i\frac{2xy+y^3+x^2y}{(1+x)^2+y^2}$$
but when I try to compute the CR and partial deravtives, I am getting extremely long and messy computations and it just doesn't seem right.
I also know that holomorphic is equivalent to having the partial deravtive with respect to z conjugate being zero. Or having a deravtive at all places.
I'm really lost and looking for advice
You have that $\sin z$ is holomorphic everywhere, and so is $z^2$. On any disk that fails to contain $-1$, the function $1/(z+1)$ is also holomorphic (one can show directly that its derivative is $-1/(z+1)^2$.
Sums and products of holomorphic functions are again holomorphic, so $f$ is holomorphic.