Evaluate tan(pi/8)

Problem

tan(π/8)

Solution

1. Identify the half-angle identity for tangent. Since π/8 is half of π/4 we use the formula:

tan(θ/2)=(1−cos(θ))/sin(θ)

2. Substitute θ=π/4 into the identity:

tan(π/8)=(1−cos(π/4))/sin(π/4)

3. Evaluate the trigonometric values for π/4

cos(π/4)=√(,2)/2

sin(π/4)=√(,2)/2

4. Simplify the expression by substituting these values:

tan(π/8)=(1−√(,2)/2)/√(,2)/2

5. Multiply the numerator and denominator by 2 to clear the fractions:

tan(π/8)=(2−√(,2))/√(,2)

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

tan(π/8)=2/√(,2)−√(,2)/√(,2)

tan(π/8)=√(,2)−1

Final Answer

tan(π/8)=√(,2)−1

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