Perhaps the two pictures below will help. The sine is positive when the height measured is to a point above the horizontal axis, and negative if below. The tangent is similarly positive or negative if the height is to a point respectively above or below the axis, but here we must clarify what happens if the point whose tangent we are taking is in the left half of the circle. You draw a line through that point and the center, and it still intersects the vertical tangent line to the right of the circle, and we go by that point. The secant is positive if the point on the circle is between the center and the intersection with that vertical tangent line; thus it is positive in the right half of the circle. In the left half of the circle, the center is between the point on the circle and the point of intersection with the vertical tangent line on the right; thus the secant is negative.
The tangent or the secant of $90^\circ$ is $\infty$, and this $\infty$ is neither $+\infty$ nor $-\infty$ but rather a single $\infty$ that is approached by going in either the positive or the negative direction along the line.