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Tangent Sum And Product

Problem

Prove that tan$A$ + tan$B$ + tan$C$ = tan$A$ tan$B$ tan$C$ 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($A$ + ($B$ + $C$)) =
tan$A$ + tan($B$ + $C$)

1 minus tan$A$ tan($B$ + $C$)
= 0

therefore tan$A$ + tan($B$ + $C$) = 0.

Using the addition formula again:

tan A +
tan$B$ + tan$C$

1 minus tan$B$ tan$C$
= 0

therefore tan$A$(1 minus tan$B$ tan$C$) + tan$B$ + tan$C$ = 0

therefore tanA minus tan$A$ tan$B$ tan$C$ + tan$B$ + tan$C$ = 0

Hence, tan$A$ + tan$B$ + tan$C$ = tan$A$ tan$B$ tan$C$

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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