Let $G$ be the relation in $\Bbb {R}^2$ given by $$ G = \{((a, b), (c, d)) \in \Bbb {R}^2 \times \Bbb {R}^2: a ^ 2 + b ^ 2 = c ^ 2 + d ^ 2 \}. $$ Prove that $ G $ is an equivalence relation. Also describe $$ f ^ * (G) = \{(x, y) \in \mathbb R^2: (f (x), f (y)) \in G \} $$ where $ f: \Bbb {R} \to \Bbb {R} ^ 2 $ is given by $ f (t) = (\sin (t), \cos (t)) $.
I already proved that $ G $ is an equivalence relation, it is quite clear and easy to do, that. But how do I describe $ f ^ * (G) $.
On
What you need is the set of pairs $(x,y)$ such that the pairs $(\sin x,\cos x)$ and $(\sin y,\cos y)$ are related. According to the definition of the relation, this happens if $\sin^2x+\cos^2x=\sin^2y+\cos^2y$. But this is always the case, since both are equal to $1$. Therefore, $f^*(G)=\mathbb{R}^2$.
For $x,y\in\Bbb R$, we have $$\begin{align} (x,y)\in f^*(G)&\iff (f(x),f(y))\in G\\&\iff \bigl((\sin x,\cos x),(\sin y,\cos y)\bigr)\in G\\&\iff \sin^2x+\cos^2x=\sin^2y+\cos^2y\end{align}$$ but the latter is true for all $x,y$ (because both sides are $=1$), hence $f^*(G)=\Bbb R\times \Bbb R$.