Define on R the relation $xTy$ if and only if $cos^2(x) + sin^2 (y) = 1$. Prove that this is an equivalence relation and find R/T
About that second part, what do the equivalence classes look like? I found that, for every real number a, $[a]_T$ = {$y: (y = 2kπ +-a)$ or $(y = 2kπ + (π-a))$ or $(y = 2kπ + (π+a))$, for all k in Z}, but I cannot describe this partition properly.
You have already (mostly) established that $$\begin{align}T&=\{\langle x,y\rangle\in\Bbb R^2\mid\cos^2(x)+\sin^2(y)=1\}\\&=\{\langle x,k\pi\pm x\rangle\mid x\in\Bbb R, k\in\Bbb Z\}\\[2ex]{[x]}_T &=\{y\in\Bbb R\mid\exists k\in\Bbb Z: y=k\pi\pm x\}&\text{for all }x\in\Bbb R \\ &=\{k\pi\pm x\mid k\in\Bbb Z\}\end{align}$$
Now, we know $\Bbb R/T = \{{[x]}_T: x\in\Bbb R\}$ by definition of Quotient Set, so...