From our knowledge of the Unit Circle, we know that tangent and cotangent is positive in the 1st quadrant (0 to π/2) and in the 3rd quadrant (π to 3π/2) and negative in the 2nd and 4th quadrants (π/2 to π and 3π/2 to 2π). Cosine is positive in the 1st and 4th quadrants, but negative in the 2nd and 3rd quadrants.So both of their graphs will be positive and negative in the same places of the graph.
However, we must remember that tangent and cotangent are different ratios.Tangent is equal to sine/cosine while cotangent is equal to cosine/sine. And from the previous concept, we know that tangent will have asymptotes whenever they are undefined, which is whenever their denominator is equal to 0. Therefore, tangent would have asymptotes whenever cosine=0, which is at π/2 and 3π/2. Of course, those are the only asymptotes in the 4 quadrants that we are looking at, but there are more asymptotes at the same intervals as well.
Now for cotangent, it is equal to cosine/sine, So it would have asymptotes whenever sine=0, which is at 0, π, and 2π.