Answer:
[tex]x = \frac{2}{3}n\pi[/tex]
Step-by-step explanation:
Given
[tex]Tanx + Tan2x + Tan3x = Tanx.Tan2x.Tan3x[/tex]
Required
Find x
Equate the given expression to 0
[tex]Tan\ x + Tan\ 2x + Tan\ 3x - Tan\ x\ .Tan\ 2x\ .Tan\ 3x = 0[/tex]
Factorize:
[tex]Tan\ x + Tan\ 2x + Tan\ 3x(1 - Tan\ x\ .Tan\ 2x) = 0[/tex]
Subtract both [tex]Tan\ 3x(1 - Tan\ x\ .Tan\ 2x)[/tex] from both sides
[tex]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)[/tex]
[tex]Tan\ x + Tan\ 2x = - Tan\ 3x(1 - Tan\ x\ .Tan\ 2x)[/tex]
Divide both sides by: [tex](1 - Tan\ x\ .Tan\ 2x)[/tex]
[tex]\frac{Tan\ x + Tan\ 2x}{(1 - Tan\ x\ .Tan\ 2x) } = - Tan\ 3x[/tex]
In trigonometry:
[tex]Tan(3x) = Tan(x + 2x) = \frac{Tan\ x + Tan\ 2x}{(1 - Tan\ x\ .Tan\ 2x) }[/tex]
So,
[tex]\frac{Tan\ x + Tan\ 2x}{(1 - Tan\ x\ .Tan\ 2x) } = - Tan\ 3x[/tex]
is equivalent to:
[tex]Tan\ 3x = - Tan\ 3x[/tex]
Collect Like Terms
[tex]Tan\ 3x+Tan\ 3x = 0[/tex]
[tex]2Tan\ 3x = 0[/tex]
Divide through by 2
[tex]Tan\ 3x = 0[/tex]
In trigonometry:
If
[tex]Tan\ x= 0[/tex] ,
Then [tex]x=2n\pi[/tex]
Where [tex]n \ge 0[/tex]
So, in this case:
[tex]3x = 2n\pi[/tex]
[tex]x = \frac{2}{3}n\pi[/tex]
