use a half - angle formula to find the exact value of the following expression. tan 105°

Question

Accepted Answer

# Explanation:
## Step1: Express $105^{\circ}$ as a half - angle
We know that $105^{\circ}=\frac{210^{\circ}}{2}$. The half - angle formula for tangent is $\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}$ (when $\frac{\alpha}{2}$ is in the second quadrant, $\tan\frac{\alpha}{2}<0$). Here $\alpha = 210^{\circ}$, $\sin210^{\circ}=-\frac{1}{2}$, $\cos210^{\circ}=-\frac{\sqrt{3}}{2}$.
## Step2: Substitute into the half - angle formula
$$
\begin{align*}
\tan105^{\circ}&=\frac{1-\cos210^{\circ}}{\sin210^{\circ}}\
&=\frac{1+\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\
&=\frac{\frac{2 + \sqrt{3}}{2}}{-\frac{1}{2}}\
&=-(2+\sqrt{3})
\end{align*}
$$

# Answer:
$-(2 + \sqrt{3})$

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Question

Explanation:

Step1: Express \(105^{\circ}\) as a half - angle

Step2: Substitute into the half - angle formula

Answer: