First, we expand the equation:
\(cos^{2}A+2cosAcosB+ cos^{2}B+sin^{2}A+2sinAsinB+sin^{2}B\)
Then, we combine certain terms in order to simplify them using trigonometry identities.
\(cos^{2}A+sin^{2}A+cos^{2}B+sin^{2}B+2cosAcosB+2sinAsinB\)
Note these identities:
Pythagoran relation: \(cos^{2}A+sin^{2}A = cos^{2}B+sin^{2}B=1\)
Sum and Difference Formula/Identities: \(cos(A-B) = cosAcosB+sinAsinB\)
Thus, if we apply these identities, the simplified equation would be:
\(2(cos(A-B))+2\)
Simplying further, the answer would be
\(2[cos(A-B)+1]\)
