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

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}))