Law of Tangents

The law of tangents states that if a triangle has sides of length , , and opposite angles , , and , then

(1)

For a nondegenerate convex cyclic quadrilateral , let , , , and , and let its interior angles at and be and , respectively. Then

(2)

(José García 2024, pp. 274-275), with the quotient interpreted by continuity when . The identity follows from the prosthaphaeresis formulas and the area identities , where is the area of the cyclic quadrilateral. The triangle identity is recovered in the limit as and coalesce on the circumcircle, so that .

An analogous result for oblique spherical triangles states that

(3)

For a hyperbolic triangle of curvature , the corresponding identity is the hyperbolic law of tangents,

(4)

Here, is the hyperbolic tangent. This identity follows from the hyperbolic law of sines by applying the prosthaphaeresis formulas (Anderson 1999, pp. 146-151).