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

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}))