We know that our Tangent value is sin/cos or y/x. Our cotangent value is x/y or cos/sin. In order to for tan and cot to go their directions we must have our asymptotes in which for tan the x value has to be 0 while for cot our y must be 0. Tangent is 0 at the degrees of 90 and 270 because we take our coordinates of our quadrant angles. 90 degrees has a coordinate of (0,1) and 270 has a point of (0,-1). Once we plug this into our equation for both we get 1/0 and -1/0 which is our asymptote. These degrees also have radians in which 90 is pi/2 and 270 is 3pi/2. These serve as our asymptotes on our graph and is represented by a dotted line. The same thing applies to cotangent in which our value is x/y in which the degrees of 0 and 180 are our asymptotes. Plugging in our values we get 1/0 and -1/0. 0 has no asymptote on the plane while 180 lies on pi.
By looking at our graph our asymptote for tan is placed at pi/2 and 3pi/2. These serve us as our walls which tan goes upwards. Tan also goes upward between 0 and pi/2. Cot goes downhill from its first asymptote at zero. Cot also goes down at 180 which lies on pi.
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