Cotangent Product Sum

Geometry Level 2

If α + β + γ = π \alpha + \beta + \gamma = \pi , what is the value of

cot ⁡ α cot ⁡ β + cot ⁡ β cot ⁡ γ + cot ⁡ γ cot ⁡ α ? \cot \alpha \cot \beta + \cot \beta \cot \gamma + \cot \gamma \cot \alpha?

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Chung Kevin by Chung Kevin , 45, Australia

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1 solution

Chung Kevin
Oct 23, 2014

In Proving Trigonometric Identites - Conditional , we showed that is α + β + γ = π \alpha + \beta + \gamma = \pi , then

tan ⁡ α + tan ⁡ β + tan ⁡ γ = tan ⁡ α tan ⁡ β tan ⁡ γ . \tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma.

Dividing throughout by tan ⁡ α tan ⁡ β tan ⁡ γ \tan \alpha \tan \beta \tan \gamma , we see that

cot ⁡ β cot ⁡ γ + cot ⁡ γ cot ⁡ α + cot ⁡ α cot ⁡ β = 1. \cot \beta \cot \gamma + \cot \gamma \cot \alpha + \cot \alpha \cot \beta = 1.

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