<s>[INST] Eliminating parameter from 2 parametric equations I am given two parametric equations as-
$ x= g\sin^2(t) - f\sin(t)\cos(t) $
$ y = f\cos^2(t) - g\sin(t)\cos(t) $
and I have to eliminate the parameter $t$.
I tried taking $\sin(t)$ common from the first equation and $\cos(t)$ from the second, and divided the two to get $ x/y = -\tan(t) $ but I couldn't proceed from there.
 [/INST] \begin{align}
\cos^2 t & = \frac 1 2 + \frac 1 2 \cos(2t) \\[10pt]
\sin^2 t & = \frac 1 2 - \frac 1 2 \cos(2t) \\[10pt]
\sin(t) \cos(t) & = \frac 1 2 \sin(2t) \\[10pt]
x & = g\sin^2 t - f\sin(t) \cos(t) = g\left( \frac 1 2 - \frac 1 2\cos(2t) \right) - f \cdot \frac 1 2 \sin(2t) \\[10pt]
y & = f \cos^2 t - g \sin(t)\cos(t) = f\left( \frac 1 2 + \frac 1 2 \cos(2t) \right) - g\cdot \frac 1 2 \sin(2t) \\[10pt]
fx - gy & = -fg\cos(2t) + \frac{g^2- f^2} 2 \sin(2t) \\[10pt]
gx - fy & = \frac{g^2-f^2} 2 - \frac {g^2+f^2} 2 \cos(2t) \\[10pt]
\sin(2t) & = \cdots \tag A \\[10pt]
\cos(2t) & = \cdots \tag B \\[10pt]
\text{Therefore } & \left( \cdots \right)^2 + \left( \cdots \right)^2 = 1. \text{ (Here insert the expressions in lines (A) and (B).)}
\end{align}
</s>