By definition, the vertical asymptote of a function is a vertical line on the coordinate plane that intersects the X-axis at a point where the value of a function is undefined and is infinitely increasing to +oo or infinitely decreasing to -oo as its argument x approaches to this point.
Since the definition of a function tan(phi) is sin(phi)/cos(phi), function tan() has asymptotes wherever cos() is equal to zero.
So, to get an asymptotes of function tan(pi x), we have to resolve an equation
cos(pi x)=0
As we know, function cos(phi) represents an abscissa (X-coordinate) of a point on a unit circle that is an endpoint of a vector at an angle phi with a positive direction of the X-axis. So, it equals to zero when this vector is either vertically directed up or down along the Y-axis, which corresponds to angles pi/2 and -pi/2. Adding periodicity, we can say that an angle must be equal to pi/2 + pi N, where N - any integer number.
At N=0 we get x=pi/2.
At N=-1 we get x=-pi/2.
At N=1 we get x=3pi/2 (same angle as -pi/2).
At N=-2 we get x=-3pi/2 (same angle as pi/2).
etc.
So, we have a solution for our equation:
pi x = pi/2 + pi N or
x = 1/2 + N,
where N - any integer number.
These values of x are those points where function tan(pi x) has vertical asymptotes.
