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

Problem

2*cos2(x)−1=0

Solution

1. Isolate the squared trigonometric term by adding 1 to both sides.

2*cos2(x)=1

2. Divide both sides by 2 to solve for the squared cosine.

cos2(x)=1/2

3. Take the square root of both sides, remembering to include both the positive and negative roots.

cos(x)=±√(,1/2)

4. Rationalize the denominator to identify standard values on the unit circle.

cos(x)=±1/√(,2)

cos(x)=±√(,2)/2

5. Determine the angles x where the cosine value is √(,2)/2 or −√(,2)/2 These occur at the midpoints of each quadrant.

x=π/4,(3*π)/4,(5*π)/4,(7*π)/4

6. Generalize the solution by noting that these points are separated by intervals of π/2

x=π/4+(n*π)/2

Final Answer

x=π/4+(n*π)/2

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