solve this by finding all solutions and the 6 closest to 0: tan^3 x + tan ^3 x - 3tanx = 3. The answers must come out to the following. All solutions: x=3pi/4+n; x=pi/3+n; Six closest to 0: [-2pi/3,-pi/3,-pi/4,pi/3,2pi/3,3pi/4]


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

solve this by finding all solutions and the 6 closest to 0: tan^3 x + tan ^3 x - 3tanx = 3. The answers must come out to the following. All solutions: x=3pi/4+n; x=pi/3+n; Six closest to 0: [-2pi/3,-pi/3,-pi/4,pi/3,2pi/3,3pi/4]

Discover how to solve the equation tan^3(x) + tan^3(x) - 3tan(x) = 3 by finding all solutions and the six values closest to 0. This problem involves advanced trigonometric and algebraic concepts, perfect for students in Grades 11-12.

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