Use the Half-angle Formulas to find the exact value of \( \tan \left(-\frac{5 \pi}{8}\right) \) \( \tan \left(-\frac{5 \pi}{8}\right)=\square \) (Simplify your answer. Type an exact answer, using radicals as needed.)

To find \( \tan \left(-\frac{5 \pi}{8}\right) \) using the Half-angle Formulas, we can rewrite \( -\frac{5\pi}{8} \) as \( -\frac{1}{2} \cdot \frac{5\pi}{4} \). We know that: \[ \tan \left(-\frac{\theta}{2}\right) = -\tan \left(\frac{\theta}{2}\right) \] If we let \( \theta = \frac{5\pi}{4} \), then we first find \( \tan \left(\frac{5\pi}{8}\right) \). The angle \( \frac{5\pi}{8} \) is half of \( \frac{5\pi}{4} \), which we can express using the angle addition formula. Now, the Half-angle Formula for tangent is: \[ \tan \left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta} \] For \( \theta = \frac{5\pi}{4} \): - \( \cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) - \( \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) Substituting these values gives: \[ \tan\left(\frac{5\pi}{8}\right) = \frac{1 - \left(-\frac{\sqrt{2}}{2}\right)}{-\frac{\sqrt{2}}{2}} = \frac{1 + \frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = \frac{2 + \sqrt{2}}{-\sqrt{2}} \] This simplifies as follows: \[ = -\frac{2 + \sqrt{2}}{\sqrt{2}} = -\frac{2}{\sqrt{2}} - 1 = -\sqrt{2} - 1 \] So, we have: \[ \tan \left(-\frac{5\pi}{8}\right) = -\tan \left(\frac{5\pi}{8}\right) = \sqrt{2} + 1 \] Therefore, the exact value of \( \tan \left(-\frac{5 \pi}{8}\right) \) is: \[ \boxed{\sqrt{2} + 1} \]