from {o, 2pie} what are the solutions of cos(squared)x- cos2x=0
To find the solutions of the equation cos^2(x) - cos(2x) = 0 over the interval from 0 to 2π, we can use algebraic manipulation and trigonometric identities.
Step 1: Express cos(2x) in terms of cos^2(x)
We know the double-angle identity for cosine, which states that cos(2x) = 2cos^2(x) - 1. By substituting this into the equation, we get:
cos^2(x) - (2cos^2(x) - 1) = 0
Step 2: Simplify the equation
Combine like terms to simplify the equation:
cos^2(x) - 2cos^2(x) + 1 = 0
- cos^2(x) + 1 = 0
Step 3: Factor the equation
Factor out the common factor to simplify the equation further:
-(cos^2(x) - 1) = 0
Step 4: Solve for cos(x)
Set each factor equal to zero and solve for cos(x):
cos^2(x) - 1 = 0
cos^2(x) = 1
cos(x) = ±1
Step 5: Find the values of x
To find the values of x within the given interval, we can substitute cos(x) with ±1 and solve for x.
When cos(x) = 1:
x = 0
When cos(x) = -1:
x = π
Therefore, the solutions to the equation cos^2(x) - cos(2x) = 0 over the interval from 0 to 2π are x = 0 and x = π.