<s>[INST] 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 $\tan A+\tan B= 8ab/((a^2+b^2)^2-4a^2)$
 [/INST] $$a^2+b^2=2+2\cos(A-B)\implies\cos(A-B)=?$$
$$ab=(\sin A+\sin B)(\cos A+\cos B)=\sin(A+B)+\frac{\sin2A+\sin2B}2$$
$$\implies ab=\sin(A+B)[1+\cos(A-B)]\implies \sin(A+B)=?$$
$$b^2-a^2=\cos2A+\cos2B+2\cos(A+B)=2\cos(A+B)[\cos(A-B)]+1$$
$$\implies\cos(A+B)=?$$
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