1. The problem is to show that the parametric equations $x = a \cos \theta$ and $y = b \sin \theta$ represent the ellipse given by the equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$\n\n2. Start with the parametric forms: $$x = a \cos \theta, \quad y = b \sin \theta.$$\n\n3. Square both expressions: $$x^2 = a^2 \cos^2 \theta, \quad y^2 = b^2 \sin^2 \theta.$$\n\n4. Divide $x^2$ by $a^2$ and $y^2$ by $b^2$: $$\frac{x^2}{a^2} = \cos^2 \theta, \quad \frac{y^2}{b^2} = \sin^2 \theta.$$\n\n5. Add these two equations: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2 \theta + \sin^2 \theta.$$\n\n6. Use the Pythagorean identity: $$\cos^2 \theta + \sin^2 \theta = 1.$$\n\n7. Therefore, $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$$ which is the equation of an ellipse.\n\nHence, the parametric equations $x = a \cos \theta$ and $y = b \sin \theta$ indeed represent the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.