The rectangular equation for the curve is [tex]y = \frac{x^{2}}{2} -1[/tex].
In this question, we have a set of parametric equation which must be reduced into a single rectangular equation both by algebraic and trigonometric means, that is to say:
[tex]x = f(\theta), y = g(\theta) \to y = h(x)[/tex]
If we know that [tex]x = 2\cdot \cos \theta[/tex] and [tex]y = \cos 2\theta[/tex], then the rectangular equation is:
[tex]y = \cos 2\theta[/tex]
[tex]y = \cos^{2}\theta - \sin^{2}\theta[/tex]
[tex]y = 2\cdot \cos^{2}\theta -1[/tex]
[tex]\cos \theta = \frac{x}{2}[/tex]
[tex]y = 2\cdot \left(\frac{x}{2} \right)^{2}-1[/tex]
[tex]y = \frac{x^{2}}{2} -1[/tex]
The rectangular equation for the curve is [tex]y = \frac{x^{2}}{2} -1[/tex].
We kindly invite to check this question on parametric equations: https://brainly.com/question/23070611
