(2sin(x)²×cos(x))²+((cos(x)²-sin(x)²)×cos(x))²

Let's simplify the given expression step by step.

The expression is:

\[
\left( 2\sin^2(x) \cdot \cos(x) \right)^2 + \left( \left( \cos^2(x) - \sin^2(x) \right) \cdot \cos(x) \right)^2
\]

### Step 1: Simplify each part of the expression.

#### First term:
\[
\left( 2\sin^2(x) \cdot \cos(x) \right)^2
\]
Simplifying this:
\[
= 4\sin^4(x) \cdot \cos^2(x)
\]

#### Second term:
\[
\left( \left( \cos^2(x) - \sin^2(x) \right) \cdot \cos(x) \right)^2
\]
We recognize that \(\cos^2(x) - \sin^2(x)\) is a standard trigonometric identity:
\[
\cos^2(x) - \sin^2(x) = \cos(2x)
\]
Thus, the second term simplifies as:
\[
\left( \cos(2x) \cdot \cos(x) \right)^2 = \cos^2(2x) \cdot \cos^2(x)
\]

### Step 2: Combine both simplified terms.

Now we have:
\[
4\sin^4(x) \cdot \cos^2(x) + \cos^2(2x) \cdot \cos^2(x)
\]

### Final Simplified Expression:

The final expression is:
\[
4\sin^4(x) \cdot \cos^2(x) + \cos^2(2x) \cdot \cos^2(x)
\]

This is the simplest form of the given trigonometric expression.

Let me know if you'd like further details or have any questions! Here are some related questions to explore:

1. What is the identity for \(\sin(2x)\) and how could it help in similar problems?
2. Can this expression be simplified further for specific values of \(x\)?
3. How does this expression behave graphically when plotted?
4. What are some other important trigonometric identities that help in simplifying expressions?
5. Can you expand this expression if you substitute values for \(x\)?

**Tip:** Always look for standard trigonometric identities (like \(\sin^2(x) + \cos^2(x) = 1\)) when simplifying expressions, as they often lead to easier computations.

This page provides a step-by-step solution for simplifying the trigonometric expression (2sin(x)²×cos(x))² + ((cos(x)² - sin(x)²)×cos(x))². Key concepts include trigonometric identities and simplification 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