Expand Using Sum/Difference Formulas tan(pi/8)

Problem

tan(π/8)

Solution

1. Identify the angle π/8 as half of a known special angle, π/4

2. Apply the formula for the half-angle of tangent, which is tan(θ/2)=(1−cos(θ))/sin(θ)

3. Substitute θ=π/4 into the formula.

tan(π/8)=(1−cos(π/4))/sin(π/4)

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

tan(π/8)=(1−√(,2)/2)/√(,2)/2

5. Simplify the complex 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 √(,2)

tan(π/8)=(2√(,2)−2)/2

7. Divide each term in the numerator by 2 to reach the final simplified form.

tan(π/8)=√(,2)−1

Final Answer

tan(π/8)=√(,2)−1

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