Simplify cos(pi/8)

Problem

cos(π/8)

Solution

1. Identify the half-angle identity for cosine, which is cos(θ/2)=±√(,(1+cos(θ))/2)

2. Substitute θ=π/4 into the identity, since π/8 is half of π/4

3. Determine the sign of the square root. Since π/8 is in the first quadrant, the cosine value must be positive.

4. Evaluate the expression using the known value cos(π/4)=√(,2)/2

cos(π/8)=√(,(1+√(,2)/2)/2)

5. Simplify the fraction inside the square root by finding a common denominator in the numerator.

cos(π/8)=√(,(2+√(,2))/2/2)

6. Divide the fractions to reach the final simplified form.

cos(π/8)=√(,(2+√(,2))/4)

7. Extract the square root of the denominator.

cos(π/8)=√(,2+√(,2))/2

Final Answer

cos(π/8)=√(,2+√(,2))/2

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