To solve the equation \(\cos(x) \cdot (\cos(x) - \sin(x)) = 0\), we'll analyze it by breaking it down into two possible cases.

### Step 1: Set Each Factor to Zero
The equation is a product of two factors:
\[
\cos(x) \cdot (\cos(x) - \sin(x)) = 0
\]
For a product to be zero, at least one of the factors must be zero.

**Case 1:** \(\cos(x) = 0\)

**Case 2:** \(\cos(x) - \sin(x) = 0\)

### Step 2: Solve Each Case

**Case 1:** \(\cos(x) = 0\)

The cosine function is zero at the following values:
\[
x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}
\]

**Case 2:** \(\cos(x) - \sin(x) = 0\)

This simplifies to:
\[
\cos(x) = \sin(x)
\]
Divide both sides by \(\cos(x)\) (assuming \(\cos(x) \neq 0\)):
\[
1 = \tan(x)
\]
\(\tan(x) = 1\) when:
\[
x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

### Final Answer
The general solution for the given equation is:
\[
x = \frac{\pi}{2} + n\pi \quad \text{or} \quad x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

Do you want more details or have any questions?

Here are some related questions:
1. What are the solutions for \(\sin(x) = 0\)?
2. How can the unit circle be used to solve trigonometric equations?
3. What is the general solution for \(\tan(x) = -1\)?
4. How do the sine and cosine functions behave over one period?
5. What is the significance of the tangent function having a period of \(\pi\)?

**Tip:** For trigonometric equations, always consider the unit circle to visualize where the sine, cosine, and tangent functions take specific values.

Explore the detailed solution to the trigonometric equation cos(x)(cos(x) − sin(x)) = 0, covering essential concepts like trigonometric identities and equation solving techniques. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation