For any two real numbers $\alpha$ and b, we define $\alpha$ R b if and only if ${\sin }^{2}a+{\cos }^{2}b=1$. The relation R is

$$\begin{gathered} {\sin }^{2}a+{\cos }^{2}b=1 \\ Reflexive: {\sin }^{2}a+{\cos }^{2}a=1\Rightarrow aRa \\ \begin{array}{l}{\sin }^{2}a+{\cos }^{2}b=1,1-{\cos }^{2}a+1-{\sin }^{2}b=1\\ {\sin }^{2}b+{\cos }^{2}a=1\Rightarrow bRa\end{array} \end{gathered}$$

Hence symmetric

Let aRb, bRc

$$\begin{gathered} {\sin }^{2}a+{\cos }^{2}b=1 \left(1\right) \\ {\sin }^{2}b+{\cos }^{2}c=1 \left(2\right) \end{gathered}$$

(i) + (ii)

${\sin }^{2}a+{\cos }^{2}c=1$