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: Substitute $\alpha = \frac{\pi}{4}$ into the formula
We know that $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.
Substitute into $\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}$, we get $\tan\frac{\pi}{8}=\frac{1-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}$.
## Step3: Simplify the expression
$$
\begin{align*}
\tan\frac{\pi}{8}&=\frac{1-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\
&=\frac{\frac{2 - \sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\
&=\frac{2-\sqrt{2}}{\sqrt{2}}\
&=\frac{(2 - \sqrt{2})\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}\
&=\frac{2\sqrt{2}-2}{2}\
&=\sqrt{2}-1
\end{align*}
$$

# Answer:
$\sqrt{2}-1$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the half - angle formula for tangent

Step2: Substitute \(\alpha = \frac{\pi}{4}\) into the formula

Step3: Simplify the expression

Answer: