\(y=A_1 \sin \left(w t-\beta_1\right)+A_2 \sin \left(w t-\beta_2\right)\)

\(= A _1 \cos \beta_1 \sin w t- A _1 \cos w t \sin \beta_1+ A _2 \sin w t \cos \beta_2- A _2 \cos w t \sin \beta_2\)

\(=\left(A_1 \cos \beta_1+A_2 \cos \beta_2\right) \sin w t-\left(A_1 \sin \beta_1+A_2 \sin \beta_2\right) \cos w t\)

Amplitude of this wave is given by

\(\sqrt{\left(A_1 \cos \beta_1+A_2 \cos \beta_2\right)^2+\left(A_1 \sin \beta_1+A_2 \sin \beta_2\right)^2}\)

\(=\sqrt{A_1^2 \cos ^2 \beta_1+A_2^2 \cos ^2 \beta_2+2 A_1 A_2 \cos \beta_1 \cos \beta_2+A_1^2 \sin ^2 \beta_1+A_2^2 \sin ^2 \beta_2+2 A_1 A_2 \sin \beta_1 \sin \beta_2}\)

\(=\sqrt{ A _1^2+ A _2^2+2 A _1 A _2 \cos \left(\beta_1-\beta_2\right)}\)