Law of cotangents

For the interior angles A, B, and C of a Euclidean triangle, the sum cot A cot B + cot B cot C + cot C cot A equals 1.

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Tangent addition formula: A trigonometric identity expressing the tangent of a sum in terms of the tangents of its angles. Applying it to A + B + C = π yields the cotangent relation.
Interior angle: An angle formed inside a polygon by two of its sides meeting at a vertex. A, B, and C denote the triangle’s interior angles.
Triangle trigonometry: The study of relations among a triangle’s sides, angles, and trigonometric functions. The law converts a constraint on three angles into an equation among their cotangents.
Law of sines: A triangle theorem stating that each side divided by the sine of its opposite angle has the same value. It relates sides to angles, whereas the cotangent law uses only the three angles.
Triangle angle sum theorem: The Euclidean geometry theorem that a triangle’s interior angles sum to 180 degrees. The identity depends on the angles satisfying A + B + C = π.
Euclidean geometry: The geometry of flat space governed by Euclid’s postulates. The stated angle sum, and therefore the identity, assumes a Euclidean triangle.
Trigonometric substitution: A method that replaces variables with trigonometric expressions to simplify equations or integrals. Cotangent substitutions can turn angle constraints into algebraic relations.
Law of cosines: A triangle theorem relating a side’s squared length to the other two sides and the cosine of their included angle. It constrains side lengths and angles rather than a sum of cotangent products.
Cotangent: The trigonometric function equal to cosine divided by sine, or the reciprocal of tangent where defined. Each term in the identity is a product of cotangents.
Right triangle: A triangle containing one angle of exactly 90 degrees. Its right angle has cotangent zero, simplifying the identity to a product of the other two cotangents equal to one.