1. **State the problem:** Convert the parametric equations $$x = 3 + 2 \sin t$$ $$y = 7 + \frac{6}{\cos 2t}$$ with parameter $$t$$ in $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$ into a Cartesian equation relating $$x$$ and $$y$$. 2. **Recall trigonometric identities:** We know that $$\sin t = \frac{x-3}{2}$$ from the first equation. Also, use the double angle identity: $$\cos 2t = 1 - 2 \sin^2 t$$. 3. **Express $$\cos 2t$$ in terms of $$x$$:** Since $$\sin t = \frac{x-3}{2}$$, $$\sin^2 t = \left(\frac{x-3}{2}\right)^2 = \frac{(x-3)^2}{4}$$. Therefore, $$\cos 2t = 1 - 2 \cdot \frac{(x-3)^2}{4} = 1 - \frac{(x-3)^2}{2} = \frac{2 - (x-3)^2}{2}$$. 4. **Rewrite $$y$$ in terms of $$x$$:** Given $$y = 7 + \frac{6}{\cos 2t} = 7 + \frac{6}{\frac{2 - (x-3)^2}{2}} = 7 + \frac{6 \cdot 2}{2 - (x-3)^2} = 7 + \frac{12}{2 - (x-3)^2}$$. 5. **Final Cartesian form:** $$y = 7 + \frac{12}{2 - (x-3)^2}$$. This is the Cartesian equation of the curve.