Question

What are the asymptotes of `cot(x+90°)`?

Answer 1

`x=k*180^"o"-90^"o"` where `k` can be any integer.

Explanation:

Recall that the cotangent function, defined as
`cot theta=(adj.)/(opp.)` in a right triangle
has vertical asymptote every time the length of the opposite side (equivalently the `y` -coordinate in a unit circle) equals zero, such that `theta` would equal to multiples of `180^"o"`.

Therefore by letting `theta=x+90^"o"`, we see that the `x` -coordinate of each asymptote shall satisfy the equation
`x+90^"o"=theta=k*180^"o"`
where `k`, as mentioned in the answer, can take any integer value (that is: `k \in ZZ`).

By solving the equation we get
`x=k*180^"o"-90^"o"`.

(This expression is equivalent to `x=k*180^"o"+90^"o"` since they differ by an integer multiple of `180^"o"` )

Graph of the `cot ((x+90)*pi/180)`
graph{cot((x+90)/(180/(pi))) [-180, 180, -5, 5]}