To solve the trigonometric equation [tex]\(\cos(x) \tan(x) - \cos(x) = 0\)[/tex], let's go through a step-by-step approach.
### Step 1: Simplify the Equation
First, look at the given equation:
[tex]\[
\cos(x) \tan(x) - \cos(x) = 0
\][/tex]
We can factor out [tex]\(\cos(x)\)[/tex] from the left-hand side:
[tex]\[
\cos(x) (\tan(x) - 1) = 0
\][/tex]
This gives us two possible solutions:
1. [tex]\(\cos(x) = 0\)[/tex]
2. [tex]\(\tan(x) - 1 = 0\)[/tex]
### Step 2: Solve [tex]\(\cos(x) = 0\)[/tex]
For [tex]\(\cos(x) = 0\)[/tex], we need to find the values of [tex]\(x\)[/tex]:
[tex]\[
\cos(x) = 0
\][/tex]
The solutions are:
[tex]\[
x = \frac{\pi}{2} + n\pi \quad \text{where } n \in \mathbb{Z}
\][/tex]
### Step 3: Solve [tex]\(\tan(x) - 1 = 0\)[/tex]
Next, solve the equation [tex]\(\tan(x) - 1 = 0\)[/tex], which simplifies to:
[tex]\[
\tan(x) = 1
\][/tex]
The solutions to [tex]\(\tan(x) = 1\)[/tex] are:
[tex]\[
x = \frac{\pi}{4} + n\pi \quad \text{where } n \in \mathbb{Z}
\][/tex]
### Step 4: Combine the Solutions
We have two sets of solutions from the two cases:
1. [tex]\(x = \frac{\pi}{2} + n\pi\)[/tex]
2. [tex]\(x = \frac{\pi}{4} + n\pi\)[/tex]
Given these, the solution to the equation [tex]\(\cos(x)\tan(x) - \cos(x) = 0\)[/tex] can be combined and simplified. Recognizing that [tex]\(x = \frac{\pi}{4}\)[/tex] is a specific solution consistent with one set of the general solutions.
The final consistent and simplified solution for the given equation is:
[tex]\[
x = \frac{\pi}{4} + n\pi \quad \text{where } n \in \mathbb{Z}
\][/tex]
Thus, the complete solution to the equation [tex]\(\cos(x) \tan(x) - \cos(x) = 0\)[/tex] is:
[tex]\[
x = \frac{\pi}{4} + 2n\pi \quad \text{for all } n \in \mathbb{Z}
\][/tex]
