Simplify 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 can use the formula:

tan(θ/2)=sin(θ)/(1+cos(θ))

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

tan(π/8)=sin(π/4)/(1+cos(π/4))

3. Evaluate the trigonometric values for π/4

sin(π/4)=√(,2)/2

cos(π/4)=√(,2)/2

4. Plug these values into the expression:

tan(π/8)=√(,2)/2/(1+√(,2)/2)

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

tan(π/8)=√(,2)/(2+√(,2))

6. Rationalize the denominator by multiplying the numerator and denominator by the conjugate 2−√(,2)

tan(π/8)=(√(,2)*(2−√(,2)))/((2+√(,2))*(2−√(,2)))

7. Expand and simplify the terms:

tan(π/8)=(2√(,2)−2)/(4−2)

tan(π/8)=(2√(,2)−2)/2

tan(π/8)=√(,2)−1

Final Answer

tan(π/8)=√(,2)−1

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