Verify the identity by simplifying the left side.
We have $$\displaystyle \sin^2x+\cos^2x=\sin^2y+\cos^2y$$ as both are equal to $1$
Now change the sides of $\displaystyle \sin^2y,\cos^2x$
This can be verified by using $\cos^2(x)=1-\sin^2(x)$ and $\cos^2(y)=1-\sin^2(y)$.
$$\sin^2 x-\sin^2 y=1-\sin^2 y-(1-\sin^2 x)=\sin^2 x-\sin^2 y,$$ because $$\cos^2 x=1-\sin^2 x,$$ and $$\cos^2 y=1-\sin^2 y$$