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

Question

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

Accepted Answer

# 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:
$\sqrt{2}-1$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the half - angle formula for tangent

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

Step3: Substitute into the half - angle formula

Answer: