Find all pairs $( x , y )$ with $x , y$ real, satisfying the equations: $$\sin \left( \frac { x + y } { 2 } \right) = 0 , \quad | x | + | y | = 1$$

For a real number $\alpha$, let $S _ { \alpha }$ denote the set of those real numbers $\beta$ that satisfy $\alpha \sin ( \beta ) = \beta \sin ( \alpha )$. Then which of the following statements is true?
(A) For any $\alpha , S _ { \alpha }$ is an infinite set.
(B) $S _ { \alpha }$ is a finite set if and only if $\alpha$ is not an integer multiple of $\pi$.
(C) There are infinitely many numbers $\alpha$ for which $S _ { \alpha }$ is the set of all real numbers.
(D) $S _ { \alpha }$ is always finite.

The locus of points ( $x , y$ ) in the plane satisfying $\sin ^ { 2 } ( x ) + \sin ^ { 2 } ( y ) = 1$ consists of
(A) A circle that is centered at the origin.
(B) infinitely many circles that are all centered at the origin.
(C) infinitely many lines with slope $\pm 1$.
(D) finitely many lines with slope $\pm 1$.

Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt { 2 } \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt { 2 } \sin \theta \}$, be two sets. Then
(1) $P \subset Q$ and $Q - P \neq \phi$
(2) $Q \not \subset P$
(3) $P = Q$
(4) $P \not \subset Q$