# Find all singularities of $\sin(1/\cos(1/z))$ and determine their type

I have the folowing problem at hand:

Find all singularities of $$\sin\left(\frac{1}{\cos\left(\frac{1}{z}\right)}\right)$$ and determine their type.

Now I believe the set of singularities are $$\left\{ \frac{\pi}{2} + n \pi \colon n \in \mathbb{Z} \right\}$$. But I find it hard to figure out what type they are from removable, poles, essential or not isolated at all. Can anyone help me?

• Welcome to MSE! A general remark that is helpful in your case is the following: If $f:\mathbb C\to \mathbb C$ is holomorphic and not a polynomial (this is oftentimes called a transcendental function), and $g$ is holomorphic with a singular point $z_0$ that is not removable, then $f\circ g$ has in $z_0$ an essential singularity. So you can reduce yourself to studying the singularities of $\cos(1/z)$. Commented Mar 30, 2020 at 11:00

Singularities are the points $$z$$ such that $$\cos (1/z)=0,$$ which occurs when $$1/z=\pi/2+n\pi$$ so that you can write the values of $$z$$ now, where $$n$$ is any integer.
Second part: We have Tylor series expansion for $$\sec z$$ by which we can get the Laurent series expansion for $$1/\cos (1/z)$$ with infinitely many terms with negative powers of $$z$$. It is very well explained over here Singularities of ${1}/{\cos(\frac{1}{z})}$
Therefore Sine function after composition has Laurent series expansion at $$z=0$$ with infinitely many terms having negative powers of $$z.$$ Hence the singularities are of essential type.