To solve the equation $\tan^3(x) + \tan^3(x) - 3\tan(x) = 3$, let's go through the steps carefully.

Step 1: Simplify the equation

The equation is: $2\tan^3(x) - 3\tan(x) = 3$ Let $u = \tan(x)$. Then the equation becomes: $2u^3 - 3u = 3$ Rearrange this to: $2u^3 - 3u - 3 = 0$

Step 2: Solve for $u$

This is a cubic equation in $u$. To find the solutions, we can try to factor or use trial and error to identify possible roots.

Trying $u = 1$:

$2(1)^3 - 3(1) - 3 = 2 - 3 - 3 = -4 \neq 0$

Trying $u = -1$:

$2(-1)^3 - 3(-1) - 3 = -2 + 3 - 3 = -2 \neq 0$

Trying $u = 3$:

$2(3)^3 - 3(3) - 3 = 2 \cdot 27 - 9 - 3 = 54 - 9 - 3 = 42 \neq 0$

Let's try using other methods like synthetic division or the Rational Root Theorem to identify possible roots, and