Given that A + B + C = 180 degrees prove that TanA + TanB + TanC = TanATanBTanC
M mathprob New member Joined Dec 3, 2006 Messages 2 Dec 12, 2006 #1 Given that A + B + C = 180 degrees prove that TanA + TanB + TanC = TanATanBTanC
S soroban Elite Member Joined Jan 28, 2005 Messages 5,586 Dec 13, 2006 #2 Hello, mathprob! This is a classic (old) problem . . . I too struggled with it and was surprised by the simple procedure . . IF we approach it correctly. Given that \(\displaystyle A\,+\,B\,+\,C\:=\:180^o\) prove that: \(\displaystyle \,\tan A\,+\,\tan B\,+\,\tan C \:= \:\tan A\cdot\tan B\cdot\tan C\) Click to expand... We have: \(\displaystyle \L\:A\,+\,B\:=\:180^o\,-\,C\) Take the tangent of both sides: \(\displaystyle \L\:\tan(A\,+\,B)\:=\:\tan(180^o\,-\,C)\) Then we have: \(\displaystyle \L\:\frac{\tan A \,+\,\tan B}{1\,-\,\tan A\cdot\tan B} \:=\:-\tan C\;\) ** Hence: \(\displaystyle \L\:\tan A\,+\,\tan B \:=\:-\tan C\,+\,\tan A\cdot\tan B\cdot\tan C\) Therefore: \(\displaystyle \L\:\tan A\,+\,\tan B\,+\,\tan C\;=\;\tan A\cdot\tan B\cdot\tan C\) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ** Two identities were used: . . \(\displaystyle \L\tan(x\,+\,y)\;=\;\frac{\tan x\,+\,\tan y}{1\,-\,\tan x\cdot\tan y}\) . . \(\displaystyle \L\tan(180^o\,-\,\theta)\;=\;-\tan\theta\)
Hello, mathprob! This is a classic (old) problem . . . I too struggled with it and was surprised by the simple procedure . . IF we approach it correctly. Given that \(\displaystyle A\,+\,B\,+\,C\:=\:180^o\) prove that: \(\displaystyle \,\tan A\,+\,\tan B\,+\,\tan C \:= \:\tan A\cdot\tan B\cdot\tan C\) Click to expand... We have: \(\displaystyle \L\:A\,+\,B\:=\:180^o\,-\,C\) Take the tangent of both sides: \(\displaystyle \L\:\tan(A\,+\,B)\:=\:\tan(180^o\,-\,C)\) Then we have: \(\displaystyle \L\:\frac{\tan A \,+\,\tan B}{1\,-\,\tan A\cdot\tan B} \:=\:-\tan C\;\) ** Hence: \(\displaystyle \L\:\tan A\,+\,\tan B \:=\:-\tan C\,+\,\tan A\cdot\tan B\cdot\tan C\) Therefore: \(\displaystyle \L\:\tan A\,+\,\tan B\,+\,\tan C\;=\;\tan A\cdot\tan B\cdot\tan C\) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ** Two identities were used: . . \(\displaystyle \L\tan(x\,+\,y)\;=\;\frac{\tan x\,+\,\tan y}{1\,-\,\tan x\cdot\tan y}\) . . \(\displaystyle \L\tan(180^o\,-\,\theta)\;=\;-\tan\theta\)
