Solve for x 2cos(4x)^2-1=0

Problem

2*cos2(4*x)−1=0

Solution

1. Isolate the trigonometric term by adding 1 to both sides and then dividing by 2.

2*cos2(4*x)=1

cos2(4*x)=1/2

2. Take the square root of both sides to solve for the cosine function, remembering to include both positive and negative roots.

cos(4*x)=±√(,1/2)

cos(4*x)=±√(,2)/2

3. Identify the reference angle where the cosine is √(,2)/2

Reference Angle=π/4

4. Determine the general solutions for 4*x in all four quadrants, which can be expressed compactly.

4*x=π/4+(n*π)/2

5. Solve for x by dividing the entire expression by 4.

x=π/16+(n*π)/8

Final Answer

x=π/16+(n*π)/8

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