Let's solve the equation:

\[
-4\cos(x) = -\sin(2x) + 1
\]

### Step 1: Use the double angle identity for \(\sin(2x)\)
Recall that:
\[
\sin(2x) = 2\sin(x)\cos(x)
\]
Substitute this into the equation:
\[
-4\cos(x) = -(2\sin(x)\cos(x)) + 1
\]
This simplifies to:
\[
-4\cos(x) = -2\sin(x)\cos(x) + 1
\]

### Step 2: Rearrange the equation
Move all terms involving \(\cos(x)\) to one side:
\[
-4\cos(x) + 2\sin(x)\cos(x) = 1
\]
Factor out \(\cos(x)\) on the left-hand side:
\[
\cos(x)(-4 + 2\sin(x)) = 1
\]

### Step 3: Solve for \(\cos(x)\)
At this point, you can either solve for \(\cos(x)\) using numerical or graphical methods, or analyze the equation further to find specific solutions for \(x\).

Do you want more detailed steps on solving it from here, or would you like help with a different approach?

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**Relative questions:**
1. How do trigonometric identities affect solving trigonometric equations?
2. What strategies can be used to solve complex trigonometric equations?
3. How does factoring help in trigonometric equations?
4. Can graphing both sides of the equation provide a solution?
5. What is the significance of the double angle identity in this context?

**Tip:** Remember that trigonometric identities like the double angle or Pythagorean identities are powerful tools for simplifying equations.

Learn how to solve the trigonometric equation -4cos(x) = -sin(2x) + 1 using the double angle identity and factoring techniques. This problem covers key trigonometric concepts and is suitable for students in Grades 10-12.

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