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

Problem

2*cos2(x)=1

Solution

1. Isolate the squared trigonometric term by dividing both sides of the equation by 2

cos2(x)=1/2

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

cos(x)=±√(,1/2)

3. Rationalize the denominator to express the value in a standard form.

cos(x)=±1/√(,2)

cos(x)=±√(,2)/2

4. Identify the angles on the unit circle where the cosine value is √(,2)/2 or −√(,2)/2 These occur at the odd multiples of π/4

x=π/4,(3*π)/4,(5*π)/4,(7*π)/4

5. Generalize the solution to include all possible rotations by adding multiples of π since the solutions are spaced exactly π/2 apart.

x=π/4+(n*π)/2

Final Answer

x=π/4+(n*π)/2

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