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LHS:
[e^A + e^-A]/2 + [e^B + e^-B]/2
RHS:
2 [e^(A+B)/2 + e^-(A+B)/2]/2 [e^(A-B)/2 + e^-(A-B)/2]/2
= 2 [e^A/2 e^B/2 + e^-A/2 e^-B/2]/2 [e^A/2 e^-B/2 + e^-A/2 e^B/2]/2
= 2 [e^A/2 e^A/2 e^B/2 e^-B/2 + e^-A/2 e^-A/2 e^-B/2 e^B/2 + e^A/2 e^-A/2 e^B/2 e^B/2 + e^-A/2 e^A/2 e^-B/2 e^-B/2]/4
= 2 [e^A + e^-A + e^B + e^-B]/4
= [e^A + e^-A]/2 + 4 [e^B + e^-B]/2
LHS == RHS
[e^A + e^-A]/2 + [e^B + e^-B]/2
RHS:
2 [e^(A+B)/2 + e^-(A+B)/2]/2 [e^(A-B)/2 + e^-(A-B)/2]/2
= 2 [e^A/2 e^B/2 + e^-A/2 e^-B/2]/2 [e^A/2 e^-B/2 + e^-A/2 e^B/2]/2
= 2 [e^A/2 e^A/2 e^B/2 e^-B/2 + e^-A/2 e^-A/2 e^-B/2 e^B/2 + e^A/2 e^-A/2 e^B/2 e^B/2 + e^-A/2 e^A/2 e^-B/2 e^-B/2]/4
= 2 [e^A + e^-A + e^B + e^-B]/4
= [e^A + e^-A]/2 + 4 [e^B + e^-B]/2
LHS == RHS
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