The half-angle identity
tan A/2 = ± √[(1 - cosA)/(1 + cos A)]
can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity
tan A/2 = sin A/(1 + cos A)
can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)

Verified step by step guidance

1. Start by using the half-angle identity \( \tan \frac{A}{2} = \pm \sqrt{\frac{1 - \cos A}{1 + \cos A}} \) to express \( \tan 22.5^\circ \).
2. Substitute \( A = 45^\circ \) into the identity, since \( 22.5^\circ = \frac{45^\circ}{2} \).
3. Calculate \( \cos 45^\circ \), which is \( \frac{\sqrt{2}}{2} \), and substitute it into the expression \( \tan 22.5^\circ = \sqrt{\frac{1 - \cos 45^\circ}{1 + \cos 45^\circ}} \).
4. Simplify the expression \( \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}} \) to show that it equals \( \sqrt{3 - 2\sqrt{2}} \).
5. Use the other half-angle identity \( \tan \frac{A}{2} = \frac{\sin A}{1 + \cos A} \) with \( A = 45^\circ \) to show that \( \tan 22.5^\circ = \sqrt{2} - 1 \), and verify that both expressions are equivalent by showing their squares are equal.

Key Concepts

Half-Angle Identities

Half-angle identities are trigonometric formulas that express the sine, cosine, and tangent of half an angle in terms of the trigonometric functions of the original angle. For tangent, the half-angle identity is given by tan(A/2) = ±√[(1 - cosA)/(1 + cosA)] or tan(A/2) = sinA/(1 + cosA). These identities are essential for simplifying expressions and solving problems involving angles that are halved.

Trigonometric Equivalence

Trigonometric equivalence refers to the property that different trigonometric expressions can yield the same value under certain conditions. In this case, the two forms of the half-angle identity for tangent can be shown to be equivalent when applied to the specific angle of 22.5°. Understanding this concept allows for the manipulation and transformation of trigonometric expressions to demonstrate their equality.

Square Root Properties

The square root property states that if a² = b², then a = b or a = -b, provided that a and b are real numbers. This property is crucial in the context of the problem, as it allows us to conclude that if two expressions for tan(22.5°) yield the same squared value, they must be equal in magnitude. This principle is used to validate the equivalence of the two different forms of the tangent half-angle identity.