(2cos^2)(x)-1=(cos^4)(x)-(sin^4)(x) Identify the erroneous 6 step(s) (those that are NOT equal to the RHS)


Let's first write down the given equation and simplify each side separately to identify any erroneous steps:

### Given Equation
\[
2\cos^2(x) - 1 = \cos^4(x) - \sin^4(x)
\]

### Step-by-Step Simplification

1. **Left-Hand Side (LHS):**
   \[
   2\cos^2(x) - 1
   \]

2. **Right-Hand Side (RHS):**
   \[
   \cos^4(x) - \sin^4(x)
   \]

3. **Simplify the RHS using the difference of squares:**
   \[
   \cos^4(x) - \sin^4(x) = (\cos^2(x))^2 - (\sin^2(x))^2 = (\cos^2(x) - \sin^2(x))(\cos^2(x) + \sin^2(x))
   \]

4. **Use the Pythagorean identity \(\cos^2(x) + \sin^2(x) = 1\):**
   \[
   (\cos^2(x) - \sin^2(x))(\cos^2(x) + \sin^2(x)) = (\cos^2(x) - \sin^2(x)) \cdot 1 = \cos^2(x) - \sin^2(x)
   \]

5. **So, the RHS simplifies to:**
   \[
   \cos^2(x) - \sin^2(x)
   \]

### Compare LHS and Simplified RHS

Now we have:
   \[
   2\cos^2(x) - 1 \quad \text{vs} \quad \cos^2(x) - \sin^2(x)
   \]

6. **Analyze the expressions:**
   \[
   2\cos^2(x) - 1 \neq \cos^2(x) - \sin^2(x)
   \]

Since the LHS \(2\cos^2(x) - 1\) is not equal to the RHS \(\cos^2(x) - \sin^2(x)\), there must be a mistake. Let's go through the erroneous steps that lead to this conclusion.

### Identify Erroneous Steps:

**Step 1:** LHS:
   \[
   2\cos^2(x) - 1
   \]

**Step 2:** RHS:
   \[
   \cos^4(x) - \sin^4(x)
   \]

**Step 3:** Use difference of squares:
   \[
   \cos^4(x) - \sin^4(x) = (\cos^2(x) - \sin^2(x))(\cos^2(x) + \sin^2(x))
   \]
This step is correct.

**Step 4:** Simplify using the Pythagorean identity:
   \[
   (\cos^2(x) - \sin^2(x))(\cos^2(x) + \sin^2(x)) = (\cos^2(x) - \sin^2(x)) \cdot 1 = \cos^2(x) - \sin^2(x)
   \]
This step is correct.

**Step 5:** Simplified RHS:
   \[
   \cos^2(x) - \sin^2(x)
   \]
This step is correct.

**Step 6:** Comparing LHS and RHS:
   \[
   2\cos^2(x) - 1 \neq \cos^2(x) - \sin^2(x)
   \]
Thus, the error lies in assuming that the initial equation \(2\cos^2(x) - 1 = \cos^4(x) - \sin^4(x)\) is correct, since:

### Summary of Erroneous Steps:
- Step 1: The left-hand side \(2\cos^2(x) - 1\) is given, but not proven to be equal to the right-hand side.
- Step 2: The right-hand side \(\cos^4(x) - \sin^4(x)\) is given, but the assumption that these are equal is incorrect.

Would you like further details on any part of this solution or any other questions? Here are some related questions you might find useful:

1. How do you simplify trigonometric expressions involving powers?
2. What is the difference of squares in algebra?
3. How do you verify trigonometric identities?
4. What are the common Pythagorean identities in trigonometry?
5. How can you rewrite trigonometric functions in terms of each other?
6. What strategies can be used to solve trigonometric equations?
7. How do you handle higher-order trigonometric functions?
8. What are some common mistakes to avoid in trigonometric simplifications?

**Tip:** Always verify each transformation step in trigonometric identities to ensure accuracy.

Understand how to identify errors in simplifying trigonometric equations with a detailed example. Explore the difference of squares formula and the Pythagorean identity. Suitable for students in Grades 10-12.

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