Question

Find all solutions of the equation in the interval [0, 2pi).

2 cos2x cosx-2 sin2x sinx = square root 2
Write your answer in radians in terms of n.

Answer 1

To solve the equation 2 cos^2(x) cosx - 2 sin^2(x) sinx = √2, we use trigonometric identities like double angle formulas to simplify and solve for x in the interval [0, 2π).

Step-by-step explanation:

We are tasked to find all solutions to the trigonometric equation 2 cos2x cosx - 2 sin2x sinx = √2 in the interval [0, 2π). To solve this, we can begin by simplifying the equation using trigonometric identities such as the double angle formulas: cos(2x) = cos2x - sin2x and sin(2x) = 2 sinx cosx. Applying these identities to the equation, we get:

2 cos(2x) cosx - 2 sin(2x) sinx = √2

Now, using the fact that cos(2x) = 2 cos2x - 1 and rearranging the terms, the equation becomes:

2(2 cos2x - 1) cosx - 4 sinx cosx sinx = √2

After further simplification, we will have an equation in terms of cosx and sinx that we can solve for x.