1. **Problem statement:** Given that angle $\theta$ terminates in the fourth quadrant and $\cos \theta = \frac{2}{5}$, find the exact value of $\tan \theta$. 2. **Recall the Pythagorean identity:** $$\sin^2 \theta + \cos^2 \theta = 1$$ This identity helps us find $\sin \theta$ when $\cos \theta$ is known. 3. **Calculate $\sin \theta$:** $$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{2}{5}\right)^2 = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}$$ $$\sin \theta = \pm \sqrt{\frac{21}{25}} = \pm \frac{\sqrt{21}}{5}$$ 4. **Determine the sign of $\sin \theta$:** Since $\theta$ is in the fourth quadrant, $\sin \theta$ is negative. Therefore, $$\sin \theta = -\frac{\sqrt{21}}{5}$$ 5. **Calculate $\tan \theta$ using the definition:** $$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{21}}{5}}{\frac{2}{5}}$$ 6. **Simplify the fraction:** $$\tan \theta = -\frac{\sqrt{21}}{5} \times \frac{5}{2} = -\frac{\sqrt{21}}{\cancel{5}} \times \frac{\cancel{5}}{2} = -\frac{\sqrt{21}}{2}$$ **Final answer:** $$\boxed{\tan \theta = -\frac{\sqrt{21}}{2}}$$