find all solutions to ($$ \tan\left(x\right)- $$$$ \sqrt3 $$)($$ \cos\left(x\right)-1)=0 $$in the interval [0,2pi)

To solve the equation 

\[
(\tan(x) - \sqrt{3})(\cos(x) - 1) = 0
\]

in the interval $[0, 2\pi)$, we need to find the values of $x$ that satisfy either $\tan(x) = \sqrt{3}$ or $\cos(x) = 1$.

### Case 1: $\tan(x) = \sqrt{3}$

The tangent function equals $\sqrt{3}$ at specific angles. We know that:

\[
\tan\left(\frac{\pi}{3}\right) = \sqrt{3}
\]

Since the tangent function has a period of $\pi$, the general solution for $x$ is:

\[
x = \frac{\pi}{3} + n\pi
\]

where $n$ is any integer. For the interval $[0, 2\pi)$, the valid solutions are:

\[
x = \frac{\pi}{3} \quad \text{and} \quad x = \frac{4\pi}{3}
\]

### Case 2: $\cos(x) = 1$

The cosine function equals 1 at specific angles. We know that:

\[
\cos(0) = 1
\]

Since the cosine function has a period of $2\pi$, the general solution for $x$ is:

\[
x = 2n\pi
\]

For the interval $[0, 2\pi)$, the only valid solution is:

\[
x = 0
\]

### Final Solutions

Combining the solutions from both cases, the solutions to the equation in the interval $[0, 2\pi)$ are:

\[
x = 0, \quad \frac{\pi}{3}, \quad \text{and} \quad \frac{4\pi}{3}
\]

These are all the solutions to the equation in the given interval.

Would you like more details on the solution process, or do you have any other questions?

Here are some related questions:

1. How do you find the general solutions to trigonometric equations?
2. What are the steps to solving a quadratic trigonometric equation?
3. How do you graph the function $\tan(x)$ and interpret its solutions?
4. What is the significance of the period in trigonometric functions?
5. How can you solve trigonometric equations using identities?

**Tip:** When solving trigonometric equations, always consider the periodic nature of the functions and check for all possible solutions within the given interval.

Learn how to solve the trigonometric equation tan(x) - sqrt(3) * (cos(x) - 1) = 0 in the interval [0, 2π). Understand the solutions involving tangent and cosine functions, including their periodic properties and specific angles. Suitable for advanced high school students.

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