若 sin(α+β)+cos(α+β)=22cos(α+π4)sinβ\sin \left( {\alpha + \beta }\right) + \cos \left( {\alpha + \beta }\right) = 2\sqrt{2}\cos \left( {\alpha + \frac{\pi }{4}}\right) \sin \betasin(α+β)+cos(α+β)=22cos(α+4π)sinβ ,则 ( )A. tan(α−β)=1\tan \left( {\alpha - \beta }\right) = 1tan(α−β)=1B. tan(α+β)=1\tan \left( {\alpha + \beta }\right) = 1tan(α+β)=1C. tan(α−β)=−1\tan \left( {\alpha - \beta }\right) = - 1tan(α−β)=−1D. tan(α+β)=−1\tan \left( {\alpha + \beta }\right) = - 1tan(α+β)=−1