Question

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

Answer 1

`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`