Trigonometric Identities, Continued

$$\begin{eqnarray*} \mr {\sin(A)-\sin(B)}{2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)}% {\cos(A)-\cos(B)}{-2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)} \mr {\sin^2(A)-\sin^2(B)}{\sin(A+B)\sin(A-B)}% {\cos^2(A)-\cos^2(B)}{-\sin(A+B)\sin(A-B)} \mrone {\cos^2(A)-\sin^2(B)}{=}{\cos(A+B)\cos(A-B)} \mrone {\tan(\theta)}{\isdef }{\frac{\sin(\theta)}{\cos(\theta)}} \mr {\tan(A)+\tan(B)}{\frac{\sin(A+B)}{\cos(A)\cos(B)}}% {\tan(A+B)}{\frac{\tan(A+B)}{1-\tan(A)\tan(B)}} \end{eqnarray*}$$