Understanding the Double Angle Formula for Cosine and Solving cos^2x = cos(2x)

cos^2x (Power to Double Angle)

To solve the equation involving cos^2x (power to double angle), we first need to understand the double angle formula for cosine

To solve the equation involving cos^2x (power to double angle), we first need to understand the double angle formula for cosine.

The double angle formula for cosine states that:

cos(2x) = cos^2(x) – sin^2(x)

Using this formula, we can rewrite the given equation as:

cos^2(x) = cos(2x)

Now, let’s solve the equation step by step.

Step 1: Use the double angle formula for cosine to rewrite the equation:
cos^2(x) = cos(2x)

Step 2: Expand the right side using the double angle formula for cosine:
cos^2(x) = cos^2(x) – sin^2(x)

Step 3: Rearrange the equation by subtracting cos^2(x) from both sides:
0 = – sin^2(x)

Step 4: Multiply both sides by -1 to get rid of the negative sign:
0 = sin^2(x)

Step 5: Take the square root of both sides:
√0 = √sin^2(x)

Simplifying, we get:
0 = sin(x)

Step 6: So the equation sin(x) = 0 indicates that x can be any angle where the sine function is equal to zero. These angles are the multiples of π (pi) since sin(0) = sin(π) = sin(2π) = 0.

So, the solutions to the equation cos^2(x) = cos(2x) are all values of x for which x = nπ, where n is an integer.

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