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#### Albert

##### Well-known member

- Jan 25, 2013

- 1,225

**Prove**

is the statement true ?

$ tan \,x + tan(x+60^o) + tan(x+120^o)= 3\, tan\, 3x$

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- Thread starter Albert
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- Thread starter
- #1

- Jan 25, 2013

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is the statement true ?

$ tan \,x + tan(x+60^o) + tan(x+120^o)= 3\, tan\, 3x$

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- Feb 7, 2012

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- Jan 25, 2013

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very good solution

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- Jan 25, 2013

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is the statement true ?

$ tan \,x + tan(x+60^o) + tan(x+120^o)= 3\, tan\, 3x----(1)$

$ tan \,x + tan(x+60^o) + tan(x+120^o)$

$ =tan \,x + tan(60^o+x) - tan(60^o-x)$

$= tan \,x + tan\,2x[(1+tan(60^o+x)\times tan(60^o-x)]$

let :$tan\, x=t$

we have:$t+\dfrac {2t}{1-t^2}\times(1+\dfrac{3-t^2}{1-3t^2})$

$=t+\dfrac{2t}{1-t^2}(\dfrac{4-4t^2}{1-3t^2})$

$=t+\dfrac{8t}{1-3t^2}$

$=\dfrac{9t-3t^3}{1-3t^2}$

$=3\,tan\,3x---(1)$

here :$1-3t^2\neq 0$

now (1) has been proved :

please use(1) and find the value of :

$tan\,1^o+tan\,5^o+tan\,9^o+------+tan\,177^o=?$

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