Now you might be thinking, what? why are there no asymptotes for sine and cosine, but asymptotes for everything else? Well the answer is pretty simple. All we need to do is take a look at the ratios for each trig function according to the unit circle.
When we look at the ratios we see that sine and cosine of theta are divided by r while everything else is divided by x or y. Now we know that on the unit circle r is equal to one so that means that sine and cosine will be divided by one. An asymptote is defined as being an undefined ratio. Since sine and cosine will never be divided by a zero they will never have an asymptote. Everything else will. Now tangent and secant will have the same asymptotes because they are both divided by x and then cosecant and cotangent will have the same asymptotes because they are both divided by y. The whole thing is as easy as r-x-y.
Reference
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