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Tangent Sum And Product

Problem

Prove that tanA + tanB + tanC = tanA tanB tanC for any non-right angle triangle.


Solution

As A + B + C = 180o, it follows that tan(A + B + C) = tan180o = 0.

Using the addition formula:

tan(A + (B + C)) =
tanA + tan(B + C)

1 minus tanA tan(B + C)
= 0

therefore tanA + tan(B + C) = 0.

Using the addition formula again:

tan A +
tanB + tanC

1 minus tanB tanC
= 0

therefore tanA(1 minus tanB tanC) + tanB + tanC = 0

therefore tanA minus tanA tanB tanC + tanB + tanC = 0

Hence, tanA + tanB + tanC = tanA tanB tanC

As A + B = 180o minus C, use tan(A + B) = tan(180o minus C) to prove the same result by a slightly simpler route.

Problem ID: 115 (Apr 2003)     Difficulty: 4 Star

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