use a half - angle formula to find the exact value of tan\frac{5\pi}{8}.
tan\frac{5\pi}{8}=\square

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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 $\alpha=\frac{5\pi}{4}$, then $\frac{\alpha}{2}=\frac{5\pi}{8}$.
## Step2: Find the values of $\sin\alpha$ and $\cos\alpha$
We know that $\sin\frac{5\pi}{4}=-\frac{\sqrt{2}}{2}$ and $\cos\frac{5\pi}{4}=-\frac{\sqrt{2}}{2}$.
## Step3: Substitute into the half - angle formula
$$
\begin{align*}
\tan\frac{5\pi}{8}&=\frac{1-\cos\frac{5\pi}{4}}{\sin\frac{5\pi}{4}}\
&=\frac{1-(-\frac{\sqrt{2}}{2})}{-\frac{\sqrt{2}}{2}}\
&=\frac{1 + \frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}\
&=\frac{\frac{2+\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}\
&=-\frac{2+\sqrt{2}}{\sqrt{2}}\
&=-\frac{(2+\sqrt{2})\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}\
&=-\frac{2\sqrt{2}+2}{2}\
&=-(1 + \sqrt{2})
\end{align*}
$$

# Answer:
$-(1+\sqrt{2})$

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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: