Tangent is sin/cos and it's asymptotes occur when cosine is equal to zero. Which would be at 90 and 270.
Cotangent
Cotangent is cos/sin. In the same as tangent, asymptotes occur when sine is zero, which happens at 0 and 180.
Secant
Secant is the inverse of sine, which is 1/sin. because sin can equal zero there will be an asymptote were sin equals zero.
Cosecant
Cosecant is inverse of cosine, which is 1/cos, Because cosine can equal zero, when it is equal to zero there will be an asymptote.
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