# Law of Tangents

The **Law of Tangents** is a rather obscure trigonometric identity that is sometimes used in place of its better-known counterparts, the law of sines and law of cosines, to calculate angles or sides in a triangle.

## Statement

If and are angles in a triangle opposite sides and respectively, then

## Proof

Let and denote , , respectively. By the Law of Sines,

(In general, since is constant in a triangle, any ratio of linear combinations applied to lengths of sides is equal to the ratio of the same linear combinations applied to the sines of the angles of the same sides.)

By the angle addition identities, as desired.

## Problems

### Introductory

*This problem has not been edited in. If you know this problem, please help us out by adding it.*

### Intermediate

In , let be a point in such that bisects . Given that , and , find .

### Olympiad

Show that .