use a half - angle identity to find the exact value of the following expression.\ntan 22.5°\nwhich of the…

Explanation:

Step1: Recall 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}).

Step2: Substitute (\alpha = 45^{\circ}) into the formula

We know that (\cos45^{\circ}=\frac{\sqrt{2}}{2}) and (\sin45^{\circ}=\frac{\sqrt{2}}{2}). Substitute these values into (\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}), we get (\tan22.5^{\circ}=\frac{1-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}).

Step3: Simplify the expression

[ \begin{align*} \tan22.5^{\circ}&=\frac{1-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\ &=\frac{\frac{2 - \sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\ &=\frac{2-\sqrt{2}}{\sqrt{2}}\ &=\frac{2}{\sqrt{2}}-\frac{\sqrt{2}}{\sqrt{2}}\ &=\sqrt{2}-1\ &=- 1+\sqrt{2} \end{align*} ]

Answer:

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