What is the angle x if tan 3x -tan^3 x=0?

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Given the equation:

tan 3x - 2tan^3 x = 0

We need to find x values.

==> tan 3x = 2tan^3 x

Now from trigonometric identities we know that:

tan 3x = (3tan x - tan^3 x)/ (1-3 tan^2 x)

Let us substitute:

==> (3tan x - tan^3 x) /...

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Given the equation:

tan 3x - 2tan^3 x = 0

We need to find x values.

==> tan 3x = 2tan^3 x

Now from trigonometric identities we know that:

tan 3x = (3tan x - tan^3 x)/ (1-3 tan^2 x)

Let us substitute:

==> (3tan x - tan^3 x) / (1-3tan^2 x) = 2tan^3 x

Now we will multiply by 1-3tan^2 x

==> 3tanx - tan^3 x = 2tan^3 x ( 1- 3tan^2 x)

==> 3tanx - tan^3 x = 2tan^3 x - 6tan^5 x

==> 6tan^5x - 3tan^3 x + 3tanx = 0

==> 3tanx(2tan^4 x - tan^2 x+1) = 0

==> Let u = tan^2 x

==> 3tanx ( 2u^2 - u +1) =0

But the equation between the brackets has no real solution.

Then the only solution is:

tanx = 0 ==> x = 0, pi, 2pi

==> x = n*pi   where n = 0, 1, 2, ....

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