Solve for x cos(2x) = square root of 2-cos(2x)

Problem

cos(2*x)=√(,2−cos(2*x))

Solution

1. Square both sides to eliminate the radical, noting that cos(2*x) must be non-negative since it equals a principal square root.

cos2(2*x)=2−cos(2*x)

2. Rearrange the equation into a standard quadratic form by moving all terms to one side.

cos2(2*x)+cos(2*x)−2=0

3. Factor the quadratic expression by treating cos(2*x) as the variable.

(cos(2*x)+2)*(cos(2*x)−1)=0

4. Solve for the trigonometric ratio by setting each factor to zero.

cos(2*x)=−2

cos(2*x)=1

5. Discard the extraneous solution cos(2*x)=−2 because the range of the cosine function is [−1,1]

cos(2*x)=1

6. Solve for the angle by identifying where the cosine function equals 1

2*x=2*π*n

7. Isolate x by dividing both sides by 2 where n is any integer.

x=π*n

Final Answer

cos(2*x)=√(,2−cos(2*x))⇒x=π*n

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