As θ runs over all real values, the point $(x,y)$ with $x=5$cosθ and $y=3$sinθ always satisfies:

Solve each equation for its ratio: $x/5=$ cosθ and $y/3=$ sinθ. Squaring and adding uses the one identity that removes θ, co$s^2$θ + si$n^2$θ = 1, giving $x^2/25+y^2/9=1$. Check at θ = 60°: $x=2.5$ and $y=2.598$, so 6.25/25 + 6.75/9 = 0.25 + 0.75 = 1. Pairing each denominator with the wrong variable gives $x^2/9+y^2/25=1$; $x^2+y^2=25$ would need both coefficients to be 5; and subtracting the squares instead of adding calls for secθ and tanθ, not cosθ and sinθ.