Laurent series of a fractional cosine and sine

Question

i was wondering how i can write the laurent series of

$$\cos(\dfrac{1}{z-2})$$ in z=2 ?

and if i want the laurent series of $$\sin(\dfrac{1}{z})$$ ( for Z=2 too , how it is because here 2 is not a singualrity and i don't know how to write it if it's not a singularity )

Answer 1

Since$$\cos(z)=1-\frac{z^2}{2!}+\frac{z^4}{4!}-\cdots,$$you have$$\cos\left(\frac1{z-2}\right)=1-\frac1{2!(z-2)^2}+\frac1{4!(z-2)^4}-\cdots$$