In trigonometry, the law of tangents or tangent rule is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a, b, and c are the lengths of the three sides of the triangle, and α, β, and γ are the angles opposite those three respective sides. The law of tangents states that
a − b a + b = tan 1 2 ( α − β ) tan 1 2 ( α + β ) . {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}.}
The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be used in any case where two sides and the included angle, or two angles and a side, are known.
Proof To prove the law of tangents one can start with the law of sines:
a sin α = b sin β = d , {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}=d,}
where d {\displaystyle d} is the diameter of the circumcircle, so that a = d sin α {\displaystyle a=d\sin \alpha } and b = d sin β {\displaystyle b=d\sin \beta } . It follows that
a − b a + b = d sin α − d sin β d sin α + d sin β = sin α − sin β sin α + sin β . {\displaystyle {\frac {a-b}{a+b}}={\frac {d\sin \alpha -d\sin \beta }{d\sin \alpha +d\sin \beta }}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}.}
Using the trigonometric identity, the factor formula for sines specifically
sin α ± sin β = 2 sin 1 2 ( α ± β ) cos 1 2 ( α ∓ β ) , {\displaystyle \sin \alpha \pm \sin \beta =2\sin {\tfrac {1}{2}}(\alpha \pm \beta )\,\cos {\tfrac {1}{2}}(\alpha \mp \beta ),}
we get
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