Question

use a half - angle identity to find the exact value of the following expression.
(\tan 22.5^{circ})
which of the following is the correct value of (\tan 22.5^{circ})?
(\bigcirc)a. (-1-sqrt{2})
(\bigcirc)b. (1 + sqrt{2})
(\bigcirc)c. (-1+sqrt{2})
(\bigcirc)d. (1-sqrt{2})

Explanation:

Step1: Apply the half - angle formula for tangent

The half - angle formula for tangent is \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\). Let \(\alpha = 45^{\circ}\), then \(\frac{\alpha}{2}=22.5^{\circ}\).
We know that \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\) and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\).
Substitute these values into the formula: \(\tan22.5^{\circ}=\frac{1-\cos45^{\circ}}{\sin45^{\circ}}\).

Step2: Simplify the expression

$$
LATEXBLOCK0
$$

Answer:

C. \(-1+\sqrt{2}\)