If $\sin A+\sin B =a,\cos A+\cos B=b$, find $\cos(A+B),\cos(A-B),\sin(A+B)$ If $\sin A+\sin B =a,\cos A+\cos B=b$, 


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*find $\cos(A+B),\cos(A-B),\sin(A+B)$

*Prove that $\tan A+\tan B= 8ab/((a^2+b^2)^2-4a^2)$

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