Expand Using Sum/Difference Formulas tan(22.5)

Problem

tan(22.5)

Solution

1. Identify the angle as a half-angle of a known value from the unit circle, specifically 22.5=45/2

2. Select the half-angle formula for tangent that is easiest to simplify:

tan(θ/2)=(1−cos(θ))/sin(θ)

3. Substitute θ=45 into the formula:

tan(22.5)=(1−cos(45))/sin(45)

4. Evaluate the trigonometric functions for 45

cos(45)=√(,2)/2

sin(45)=√(,2)/2

5. Substitute these values back into the expression:

tan(22.5)=(1−√(,2)/2)/√(,2)/2

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

tan(22.5)=(2−√(,2))/√(,2)

7. Rationalize the denominator by multiplying the numerator and denominator by √(,2)

tan(22.5)=((2−√(,2))√(,2))/2

tan(22.5)=(2√(,2)−2)/2

8. Divide each term in the numerator by 2

tan(22.5)=√(,2)−1

Final Answer

tan(22.5)=√(,2)−1

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