complex analysis - Laurent Series for singularities and poles

Hi guys I was wondering how I can understand if the sin and the cos has essential singularities. for instance if I want to understand if 0 which singularity is i, can write the Laurent series only of the sin (centred in 0) and see how it works , or MUST write the Laurent series of all the function (centered in zero) ?

Same for cos , help I want to understand this topic very well. Thk.

$$\int_{+\partial D}\dfrac{\sin\left(\dfrac{1}{z}\right)\cos\left(\dfrac{1}{z-2}\right)}{z-5}\,\mathrm{dz}$$