Law of tangents

In trigonometry, the tangent set ( also Rule of Napier ) establishes a relationship between the three sides of a plane triangle and the tangent of half the sum and half the difference of two angles of the triangle.

For the three pages A, B and C of a triangle, and for those pages respectively opposite angles α, β and γ where:

Because we can also write this formula as

Analogous formulas for and are obtained by cyclic permutation.

Evidence

It follows from the law of sines

And from the addition theorems obtained

As well as

Division yields the desired formula.