we have
[tex]\begin{gathered} x=4\cos t \\ \frac{x}{4}=\cos t \end{gathered}[/tex][tex]\begin{gathered} y=5\sin t \\ \frac{y}{5}=\sin t \end{gathered}[/tex]
squared both sides each equation
[tex]\begin{gathered} \frac{x^2}{4^2}=\cos ^2t \\ \\ \frac{y^2}{5^2}=\sin ^2t \end{gathered}[/tex]
Adds both equations
[tex]\frac{x^2}{16}+\frac{y^2}{25}=\cos ^2t+\sin ^2t[/tex]
Remember that
[tex]\cos ^2t+\sin ^2t=1[/tex]
so
[tex]\frac{x^2}{16}+\frac{y^2}{25}=1[/tex]
This is the equation of a vertical ellipse
see the figure below
The equation of a vertical ellipse with center at origin is equal to
[tex]\frac{x^2}{b^2}+\frac{y^2}{a^2}=1[/tex]
where
a=5
b=4
center (0,0)
