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1) Tanx + Tan2x + Tan3x = Tanx.Tan2x.Tan3x
3​

User Maca
by
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1 Answer

3 votes

Answer:


x = (2)/(3)n\pi

Explanation:

Given


Tanx + Tan2x + Tan3x = Tanx.Tan2x.Tan3x

Required

Find x

Equate the given expression to 0


Tan\ x + Tan\ 2x + Tan\ 3x - Tan\ x\ .Tan\ 2x\ .Tan\ 3x = 0

Factorize:


Tan\ x + Tan\ 2x + Tan\ 3x(1 - Tan\ x\ .Tan\ 2x) = 0

Subtract both
Tan\ 3x(1 - Tan\ x\ .Tan\ 2x) from both sides


Tan\ x + Tan\ 2x + Tan\ 3x(1 - Tan\ x\ .Tan\ 2x) - Tan\ 3x(1 - Tan\ x\ .Tan\ 2x) = 0 - Tan\ 3x(1 - Tan\ x\ .Tan\ 2x)


Tan\ x + Tan\ 2x = - Tan\ 3x(1 - Tan\ x\ .Tan\ 2x)

Divide both sides by:
(1 - Tan\ x\ .Tan\ 2x)


(Tan\ x + Tan\ 2x)/((1 - Tan\ x\ .Tan\ 2x) ) = - Tan\ 3x

In trigonometry:


Tan(3x) = Tan(x + 2x) = (Tan\ x + Tan\ 2x)/((1 - Tan\ x\ .Tan\ 2x) )

So,


(Tan\ x + Tan\ 2x)/((1 - Tan\ x\ .Tan\ 2x) ) = - Tan\ 3x

is equivalent to:


Tan\ 3x = - Tan\ 3x

Collect Like Terms


Tan\ 3x+Tan\ 3x = 0


2Tan\ 3x = 0

Divide through by 2


Tan\ 3x = 0

In trigonometry:

If


Tan\ x= 0 ,

Then
x=2n\pi

Where
n \ge 0

So, in this case:


3x = 2n\pi


x = (2)/(3)n\pi

User Mert Celik
by
7.1k points