To solve the equation [tex]\(\tan(x)(\tan(x) + 1) = 0\)[/tex], we look for the values of [tex]\(x\)[/tex] that satisfy the equation.
First, let's break the original equation into its factors:
[tex]\[
\tan(x) \cdot (\tan(x) + 1) = 0
\][/tex]
For this product to be zero, at least one of the factors must be zero. This gives us two separate equations to solve:
1. [tex]\(\tan(x) = 0\)[/tex]
2. [tex]\(\tan(x) + 1 = 0\)[/tex]
### Solving [tex]\(\tan(x) = 0\)[/tex]
The tangent function [tex]\(\tan(x)\)[/tex] is zero at integer multiples of [tex]\(\pi\)[/tex]:
[tex]\[
x = n\pi \quad \text{for any integer } n.
\][/tex]
### Solving [tex]\(\tan(x) + 1 = 0\)[/tex]
Rewriting the equation, we get:
[tex]\[
\tan(x) = -1
\][/tex]
The tangent function [tex]\(\tan(x)\)[/tex] equals [tex]\(-1\)[/tex] at angles where [tex]\(x\)[/tex] is an odd multiple of [tex]\(\frac{\pi}{4}\)[/tex]. Therefore:
[tex]\[
x = \frac{3\pi}{4} + n\pi \quad \text{for any integer } n.
\][/tex]
Combining both solutions, we obtain:
[tex]\[
x = n\pi \quad \text{and} \quad x = \frac{3\pi}{4} + n\pi \quad \text{for any integer } n.
\][/tex]
Now we match these combined solutions with the given answer choices:
[tex]\[
\text{A. } x= \pm n\pi, \quad x=\frac{\pi}{4} \pm n\pi
\][/tex]
[tex]\[
\text{B. } x=\frac{\pi}{3} \pm 2 \pi n, \quad x=\frac{3 \pi}{4} \pm 2 \pi n
\][/tex]
[tex]\[
\text{C. } x=\pm \pi n, \quad x=\frac{\pi}{2} \pm 2 \pi n
\][/tex]
[tex]\[
\text{D. } x= \pm \pi n, \quad x=\frac{3 \pi}{4} \pm \pi n
\][/tex]
The combined solutions [tex]\(x = n\pi \quad \text{and} \quad x = \frac{3\pi}{4} + n\pi\)[/tex] correspond to choice D:
[tex]\[
\text{D. } x= \pm \pi n, x=\frac{3 \pi}{4} \pm \pi n
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
