Simplify tan(22.5)

Problem

tan(22.5)

Solution

1. Identify the angle as a half-angle of a known special angle, since 22.5=45/2

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

3. Substitute θ=45 into the formula.

tan(22.5)=sin(45)/(1+cos(45))

4. Evaluate the trigonometric functions for 45 where sin(45)=√(,2)/2 and cos(45)=√(,2)/2

tan(22.5)=√(,2)/2/(1+√(,2)/2)

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

tan(22.5)=√(,2)/(2+√(,2))

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

tan(22.5)=(√(,2)*(2−√(,2)))/((2+√(,2))*(2−√(,2)))

7. Expand the terms in the numerator and denominator.

tan(22.5)=(2√(,2)−2)/(4−2)

8. Divide the terms by the denominator 2

tan(22.5)=(2√(,2)−2)/2

tan(22.5)=√(,2)−1

Final Answer

tan(22.5)=√(,2)−1

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