1. **State the problem:** Solve the differential equation $ (1 + y \tan(x)) \, dy - (1 + \cos(x)) \, dx = 0 $. 2. **Rewrite the equation:** Rearrange terms to isolate $dy$ and $dx$: $$ (1 + y \tan(x)) \, dy = (1 + \cos(x)) \, dx $$ 3. **Express as a separable form:** $$ \frac{dy}{dx} = \frac{1 + \cos(x)}{1 + y \tan(x)} $$ 4. **Check if separable:** The right side is not easily separable due to $y \tan(x)$ in the denominator. Try substitution or rearrangement. 5. **Rewrite the original equation:** $$ (1 + y \tan(x)) \, dy = (1 + \cos(x)) \, dx $$ 6. **Divide both sides by $(1 + y \tan(x))(1 + \cos(x))$ to check exactness or integrate factor:** This is complicated; instead, try to write in differential form: $$ M(x,y) = -(1 + \cos(x)), \quad N(x,y) = 1 + y \tan(x) $$ 7. **Check exactness:** $$ \frac{\partial M}{\partial y} = 0, \quad \frac{\partial N}{\partial x} = y \sec^2(x) $$ Since $\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$, not exact. 8. **Try integrating factor depending on $x$:** $$ \mu(x) = e^{\int \frac{\partial M/\partial y - \partial N/\partial x}{N} dx} $$ Calculate numerator: $$ 0 - y \sec^2(x) = - y \sec^2(x) $$ Divide by $N = 1 + y \tan(x)$: $$ \frac{- y \sec^2(x)}{1 + y \tan(x)} $$ This depends on $y$, so integrating factor depending only on $x$ unlikely. 9. **Try integrating factor depending on $y$:** $$ \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = y \sec^2(x) - 0 = y \sec^2(x) $$ Divide by $M = -(1 + \cos(x))$: $$ \frac{y \sec^2(x)}{-(1 + \cos(x))} = - y \frac{\sec^2(x)}{1 + \cos(x)} $$ Depends on $x$, so integrating factor depending only on $y$ unlikely. 10. **Try substitution:** Let $u = 1 + y \tan(x)$, then $$ du = dy \tan(x) + y \sec^2(x) dx $$ Rewrite original equation in terms of $u$ and $x$ to simplify. 11. **Alternatively, solve implicitly:** Rearrange original equation: $$ (1 + y \tan(x)) dy = (1 + \cos(x)) dx $$ Integrate both sides: $$ \int (1 + y \tan(x)) dy = \int (1 + \cos(x)) dx $$ But $y$ and $x$ mixed, so direct integration not possible. 12. **Conclusion:** The equation is nonlinear and not exact; substitution or numerical methods may be needed. For this problem, the implicit solution is given by integrating factor or substitution methods beyond this scope. **Final answer:** The differential equation is not separable or exact in simple form; advanced methods or substitutions are required for solution.