Find the Exact Value cot(pi/8)

Problem

cot(π/8)

Solution

1. Identify the half-angle relationship, noting that π/8 is half of π/4

2. Apply the formula for the cotangent half-angle, which is cot(θ/2)=(1+cos(θ))/sin(θ)

3. Substitute θ=π/4 into the formula.

cot(π/8)=(1+cos(π/4))/sin(π/4)

4. Evaluate the trigonometric functions at π/4 where cos(π/4)=√(,2)/2 and sin(π/4)=√(,2)/2

cot(π/8)=(1+√(,2)/2)/√(,2)/2

5. Simplify the complex fraction by multiplying both the numerator and the denominator by 2

cot(π/8)=(2+√(,2))/√(,2)

6. Rationalize the denominator by dividing each term in the numerator by √(,2)

cot(π/8)=2/√(,2)+√(,2)/√(,2)

cot(π/8)=√(,2)+1

Final Answer

cot(π/8)=√(,2)+1

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