Law of tangents
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In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane.
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
![\frac{a-b}{a+b} = \frac{\tan[\frac{1}{2}(\alpha-\beta)]}{\tan[\frac{1}{2}(\alpha+\beta)]}.](../../../math/1/4/e/14e5740e715d618d2f6f6f2b3e774900.png)
The law of tangents, although not as commonly known as the law of sines or the law of cosines, is just as useful, and can be used in any case where you know either two sides and an angle, or two angles and a side.
[edit] Proof
To prove the law of tangents we start with the law of sines:
This implies that
Using trigonometric identities we can say
Thus we have
Dividing both sides by 
we attain
Cross-dividing shows that
Hence the law of tangents:
