In the Unit Circle, tangent and cotangent are both positive in the first quadrant and negative in the second. Then it is positive in the third and then negative in the fourth quadrant. The identity for tangent is tan(x)=sin(x)/cos(x) and the identity for cotangent is cot(x)= cos(x)/sin(x). This causes the asmyptotes to be place in different positions. The asymptotes for tangent are in different places than the asmyptotes for cotangent and this is what makes the trig graph go uphill or downhill.
The reason why tangent is going uphill and it is normal is because the first asymptote is place at pi/2 which is the end of the first quadrant. In the first quadrant the function needs to be going up, in order to go on to the next part of the function it needs to start down without touching the asymptote. That is why it starts over and starts down then goes up this is why it is going uphill.
Cotangent is different. The first asymptote is mark at pi. This means the function can start up in quadrant one and progress down to quadrant two. There is no need for a new function to start because there is no asymptote separating those two quadrants. This is why it is normal for the function to be going down hill.
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