Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $

\begin{eqnarray} \sin(A+B)\sin(A-B) &=& (\sin A\cos B+\cos A\sin B)(\sin A\cos B-\cos A\sin B)\\\\ &=& \sin^2 A \cos^2 B -\sin^2 B \cos^2 A\\\\ &=& \sin^2 A \cos^2 B -\sin^2 B (1-\sin^2 A)\\\\ &=& \sin^2 A (\cos^2 B + \sin^2 B) - \sin^2 B\\\\ &=& \sin^2 A - \sin^2 B \end{eqnarray}

$$ \begin{align*}\sin (A+B)\cdot\sin (A-B)&=\frac{\cos(2B)-\cos (2A)}{2}\\&=\frac{(1-2\sin^2B)-(1-2\sin^2A)}{2}\\&=\sin^2A-\sin^2B\end{align*}$$

Here i have used $$\sin x\cdot\sin y=\frac{\cos(x-y)-\cos(x+y)}{2}$$

Hint: $(a+b)(a-b)=a^2-b^2$.
Then use $\sin^2\theta+\cos^2\theta=1$

Use this formula:

$$2 \sin(A+B)\sin(A-B)=\cos2B-\cos2A$$

It will be like this:

$$\dfrac12 \cdot (\cos2B-\cos2A)$$ $$=\dfrac {(1-2\sin^2B)-(1-2\sin^2A)}{2}$$

It will give the answer if you simplify.