Tangent? tanx=sinx/cosx. if sine and cosine are positive, tangent is also positive. If one of them is negative, tangent is negative. If both are negative, obviously tangent is positive. Therefore there would be asymptotes where cosx=0, so where x is equal to zero. There would be asymptotes at pi/2 and also at 3pi/2 and it would go on forever.
Cotangent? cotx=cosx/sinx. The ratio of cosx/sinx because of triangles would be x/y . Therefore, there would be asymptotes where y is equal to 0 which would be at 0 and pi.\
Secant? Since secant is the reciprocal of cosine, you would have an asymptote where 1/x this will be undefined, so where its x=0 on the unit circle which would be 0 and pi.
Cosecant? Cosecant is the reciprocal of sine, 1/y. There would be asymptotes where y=0 because i-1/0 is 0. y=0 so there would be asymptotes at pi/2 and 3pi/2 and will go on forever.
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