1. **State the problem:** Evaluate the integral $$\int e^{\tan\theta} (\sec\theta - \sin\theta) \, d\theta.$$\n\n2. **Recall relevant formulas and rules:** We will use substitution and properties of derivatives of trigonometric functions. Note that $$\frac{d}{d\theta}(\tan\theta) = \sec^2\theta$$ and $$\frac{d}{d\theta}(e^{\tan\theta}) = e^{\tan\theta} \sec^2\theta.$$\n\n3. **Rewrite the integral:** $$\int e^{\tan\theta} (\sec\theta - \sin\theta) \, d\theta = \int e^{\tan\theta} \sec\theta \, d\theta - \int e^{\tan\theta} \sin\theta \, d\theta.$$\n\n4. **Focus on the first integral:** Let $$I_1 = \int e^{\tan\theta} \sec\theta \, d\theta.$$\n\n5. **Focus on the second integral:** Let $$I_2 = \int e^{\tan\theta} \sin\theta \, d\theta.$$\n\n6. **Try substitution for $I_1$:** Let $$u = \tan\theta,$$ then $$du = \sec^2\theta \, d\theta,$$ so $$d\theta = \frac{du}{\sec^2\theta}.$$\n\nSubstitute into $I_1$:\n$$I_1 = \int e^u \sec\theta \frac{du}{\sec^2\theta} = \int e^u \frac{1}{\sec\theta} du = \int e^u \cos\theta \, du.$$\n\nBut $\cos\theta$ is not expressed in terms of $u$, so this substitution is complicated.\n\n7. **Try substitution for $I_2$:** Similarly, for $I_2$, no straightforward substitution appears.\n\n8. **Rewrite the original integrand:** Note that $$\sec\theta - \sin\theta = \frac{1}{\cos\theta} - \sin\theta = \frac{1 - \sin\theta \cos\theta}{\cos\theta}.$$ This does not simplify nicely.\n\n9. **Try differentiating $e^{\tan\theta} \cos\theta$:**\n$$\frac{d}{d\theta} \left(e^{\tan\theta} \cos\theta\right) = e^{\tan\theta} \sec^2\theta \cos\theta + e^{\tan\theta} (-\sin\theta) = e^{\tan\theta} (\sec^2\theta \cos\theta - \sin\theta).$$\n\nSince $$\sec^2\theta \cos\theta = \frac{1}{\cos^2\theta} \cos\theta = \frac{1}{\cos\theta} = \sec\theta,$$\nwe have\n$$\frac{d}{d\theta} \left(e^{\tan\theta} \cos\theta\right) = e^{\tan\theta} (\sec\theta - \sin\theta).$$\n\n10. **Therefore, the integral is:**\n$$\int e^{\tan\theta} (\sec\theta - \sin\theta) \, d\theta = e^{\tan\theta} \cos\theta + C,$$ where $C$ is the constant of integration.\n\n**Final answer:** $$\boxed{e^{\tan\theta} \cos\theta + C}.$$