The picture about shows both the tangent and cotangent graphs. The red graph represents a tangent graph while the blue graph shows a cotangent graph. As we can see, both graphs are going in opposite directions. This is because both graphs have different asymptotes which makes the direction of the graphs different. We know that tangent has a trig ratio of sine/cosine. If we make cosine equal 0, the asymptotes will lie on 90 and 270 degrees. However, using the trig ratio of cotangent, sine of 0 will land on 0 and 180 degrees. Thus, it makes the asymptotes lie on different locations causing a the directions to be opposite.
References:
http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg97.gif
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