To solve the equation $\tan(x) - \cot(x) = 2\sqrt{3}$, we can follow these steps:

Step 1: Express $\cot(x)$ in terms of $\tan(x)$

We know that: $\cot(x) = \frac{1}{\tan(x)}$ So, the equation becomes: $\tan(x) - \frac{1}{\tan(x)} = 2\sqrt{3}$

Step 2: Multiply the entire equation by $\tan(x)$ to eliminate the fraction

$\tan^2(x) - 1 = 2\sqrt{3} \cdot \tan(x)$

Step 3: Rearrange the equation to form a quadratic equation

$\tan^2(x) - 2\sqrt{3} \cdot \tan(x) - 1 = 0$

Step 4: Solve the quadratic equation using the quadratic formula

The quadratic formula is given by: $\tan(x) = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -2\sqrt{3}$, and $c = -1$.

Substitute these values into the formula: $\tan(x) = \frac{2\sqrt{3} \pm \sqrt{(2\sqrt{3})^2 + 4}}{2}$

Step 5: Simplify under the square root

$\tan(x) = \frac{2\sqrt{3} \pm \sqrt{12 + 4}}{2}$ $\tan(x) = \frac{2\sqrt{3} \pm \sqrt{16}}{2}$ $\tan(x) = \frac{2\sqrt{3} \pm 4}{2}$

Step 6: Solve for the two possible values of $\tan(x)$

1. $\tan(x) = \frac{2\sqrt{3} + 4}{2} = \sqrt{3} + 2$
2. $\tan(x) = \frac{2\sqrt{3} - 4}{2} = \sqrt{3} - 2$

Step 7: Find the corresponding angles $x$

We now have two equations: $\tan(x) = \sqrt{3} + 2$ $\tan(x) = \sqrt{3} - 2$ These values for $\tan(x)$ do not correspond to standard angles, so $x$ can be found using the arctangent (inverse tangent) function: $x = \tan^{-1}(\sqrt{3} + 2)$ $x = \tan^{-1}(\sqrt{3} - 2)$ These solutions will give the principal values of $x$, but since the tangent function is periodic with period $\pi$, the general solutions will be: $x = \tan^{-1}(\sqrt{3} + 2) + n\pi$ $x = \tan^{-1}(\sqrt{3} - 2) + n\pi$ where $n$ is any integer.

Would you like further details on any of these steps? Here are some related questions to consider:

1. What is the significance of the quadratic formula in solving equations involving trigonometric functions?
2. How does the periodicity of the tangent function affect the general solution?
3. Can the solutions be simplified or expressed in a different trigonometric form?
4. What are the properties of the tangent and cotangent functions?
5. How would this equation change if the original equation had been $\tan(x) + \cot(x) = 2\sqrt{3}$?
6. What are the implications of having a non-standard angle for tangent?
7. How would the solution differ if the right-hand side of the equation were negative?
8. Can this problem be solved graphically or using a different method?

Tip: When solving trigonometric equations, always consider the periodic nature of the functions involved to find all possible solutions.