Construct a supplementary spherical angle with apex and sides , , . Let , , be the vertices and , , be the plane angles at those vertices. Then:

,

,

,

and

,

,

.

These are the cosine rules for the sides of a spherical triangle:

,

,

.

Note that and . Apply the cosine rules for the sides on the supplementary angle:

,

,

.

In the same way,

,

.

The last three identities are known as the cosine rules for the angles or the second cosine theorem for a spherical angle.