Law of Tangents

The Law of Tangents is a useful trigonometric identity that, along with the law of sines and law of cosines, can be used to determine angles in a triangle. Note that the law of tangents usually cannot determine sides, since only angles are involved in its statement.

Theorem

The law of tangents states that in triangle $\triangle ABC$ , if $A$ and $B$ are angles of the triangle opposite sides $a$ and $b$ respectively, then $\frac{a-b}{a+b}=\frac{\tan (A-B)/2}{\tan (A+B)/2}$ .

Proof

First we can write the RHS in terms of sines and cosines: $\frac{a-b}{a+b}=\frac{\sin (A-B)/2 \cos (A+B)/2}{\sin (A+B)/2\cos (A-B)/2}$ We can use various sum-of-angle trigonometric identities to get: $\frac{a-b}{a+b}=\frac{\sin A-\sin B}{\sin A +\sin B}$ By the law of sines, we have $\frac{a-b}{a+b}=\frac{a/2R-b/2R}{a/2R+b/2R}$ where $R$ is the circumradius of the triangle. Applying the law of sines again, $\frac{a-b}{a+b}=\frac{\tan (A-B)/2}{\tan (A+B)/2}$ as desired. ∎