use a half - angle formula to find the exact value of tan (\frac{pi}{8}).\n(\tan\frac{pi}{8}=square)

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=\frac{\pi}{4}), then (\frac{\alpha}{2}=\frac{\pi}{8}).

Step2: Find the values of (\sin\alpha) and (\cos\alpha)

We know that (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}) and (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}).

Step3: Substitute into the half - angle formula

[ \begin{align*} \tan\frac{\pi}{8}&=\frac{1-\cos\frac{\pi}{4}}{\sin\frac{\pi}{4}}\ &=\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})\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}\ &=\frac{2\sqrt{2}-2}{2}\ &=\sqrt{2}-1 \end{align*} ]

Answer:

(\sqrt{2}-1)