What is $\sin^2(x) -\cos^2(y)$? For 4.3.40 to 4.3.42 of my copy of abramowitz and stegun, the relationships between squares of sines and cosines was discussed. It provide the following formulas:
$$\sin^2(x)-\sin^2(y)=\sin(x + y)\sin(x - y)$$
   $$\cos^2(x)-\cos^2(y)=-\sin(x + y)\sin(x - y)$$
   $$\cos^2(x)-\sin^2(y)=\cos(x + y)\cos(x - y)$$
What is for $\sin^2(x)-\cos^2(y)$? And how did these formulas have been derived?
 A: Hint: Use the identity 
$$ \sin^2 x+\cos^2 x =1 .$$
A: $\sin^2(x) - \cos^2(y) = - (\cos^2(y)-\sin^2(x)) = \ ?$
A: The first question is answered by ronno. And, for your second question, you can see that these are true just by simplifying the right hand side of each equation.
A: If one accepts these three identities:
$$
\sin^2\theta + \cos^2\theta=1
$$
$$
\sin(x+y)=\sin x \cos y + \cos x \sin y
$$
$$
\cos(x+y)=\cos x \cos y - \sin x \sin y
$$
Then a large class of other identities follows, including the ones in your question.
Now why would a person accept the above three identities? I don't know of their historical proofs although the first is usually attributed to pythagoras. The way one would go about proving these identities depends on the way one $\textbf{defines}$ the concepts of $\sin$ and $\cos$. 
My preferred definition involves infinite series and the methods I would use to prove these identities rely on ideas from analysis (which includes calculus) and the use of complex numbers. This is most definitely not the way that these identities were historically proved but I feel it to be a more fundamental way of understanding them. 
