Q.Prove that $\sin(x+y)\sin(x-y) = \sin^2x - \sin^2y$.

The formulas $\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y$

and $\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y$ express the sine and cosine of a sum or

difference of two angles in terms of the sines and cosines of the individual angles;

$\cos(x-y)$ is established first, directly from the unit circle using the distance between two

points on it, and every other case -- $\cos(x+y)$, $\sin(x+y)$, $\sin(x-y)$, and the

corresponding formulas for $\tan(x\pm y)$ and $\cot(x\pm y)$ -- is deduced from it. A companion

family, the sum-to-product formulas ($\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$ and its

three relatives), rewrites a sum or difference of two trigonometric terms as a product, which is

exactly what is needed to simplify an expression, prove an identity, or factor an equation

before solving it. Together these formulas are the single most-used toolkit in the chapter,

underlying the multiple-angle formulas, many identity proofs, and the standard-angle values such

as $\sin75^\circ$ or $\tan15^\circ$ that are not directly read off the unit circle.