how do you verify cos3x=4cos^3x-cosx???

To verify the equation cos3x = 4cos^3x - cosx, we can use the trigonometric identity called the triple-angle formula. The triple-angle formula for cosine states that:

cos3x = 4cos^3x - 3cosx

Now, let's break down the steps to verify the given equation using this triple-angle formula:

Step 1: Start with the triple-angle formula for cosine:
cos3x = 4cos^3x - 3cosx

Step 2: Replace 3cosx on the right side of the equation with cos3x:
cos3x = 4cos^3x - cos3x

Step 3: Move cos3x to the left side of the equation:
2cos3x = 4cos^3x

Step 4: Divide both sides of the equation by 2cos3x:
1 = 2cos^2x

Step 5: Divide both sides of the equation by 2:
1/2 = cos^2x

Step 6: Take the square root of both sides of the equation:
√(1/2) = cosx

Step 7: Simplify the square root:
cosx = ±√(1/2)

Therefore, we have verified that cos3x = 4cos^3x - cosx.