In analyzing the spherical pendulum the cotangent space of the sphere is defined as

$ T^*S^2 = \lbrace (q,p) \in \mathbb{R}^3 \times \mathbb{R}^3; |q| = 1, q \cdot p = 0 \rbrace$

my problem with this is that I see the right-hand side of the equation as a set of points, whereas I see the left-hand side as a set of linear functions on the tangent space of $S^2$.

How can I see them as the same?