The claim in 6 papers was that the expressions yield identical valures for arbitrary reals a0, b0, a1, b1 and x.

{a0 Sin[x+b0] + a1 Sin[x+b1],(Sqrt[(((a0 Cos[b0]+a1 Cos[b1]))^2)+(((a0 Sin[b0]+a1 Sin[b1]))^2)])Sin[x+ArcTan[((a0 Sin[b0]+a1 Sin[b1])/(a0 Cos[b0]+a1 Cos[b1]))]]}

That above claim is not true can be seen by subsitituting particular values

{a0 Sin[x+b0] + a1 Sin[x+b1],(Sqrt[(((a0 Cos[b0]+a1 Cos[b1]))^2)+(((a0 Sin[b0]+a1 Sin[b1]))^2)])Sin[x+ArcTan[((a0 Sin[b0]+a1 Sin[b1])/(a0 Cos[b0]+a1 Cos[b1]))]]}/.{a0-> -1,a1-> -1,b0-> 0,b1-> 0}

It is also clear that the second expression is not defined for all reals a0, b0, a1, b1 and x:

{a0 Sin[x+b0] + a1 Sin[x+b1],(Sqrt[(((a0 Cos[b0]+a1 Cos[b1]))^2)+(((a0 Sin[b0]+a1 Sin[b1]))^2)])Sin[x+ArcTan[((a0 Sin[b0]+a1 Sin[b1])/(a0 Cos[b0]+a1 Cos[b1]))]]}/.{a0-> 1,a1-> 1,b0-> Pi/2,b1-> Pi/2}

To explore these two expressions further we can use Manipulate:

Manipulate[
Plot[{a0 Sin[x+b0] + a1 Sin[x+b1],(Sqrt[(((a0 Cos[b0]+a1 Cos[b1]))^2)+(((a0 Sin[b0]+a1 Sin[b1]))^2)])Sin[x+ArcTan[((a0 Sin[b0]+a1 Sin[b1])/(a0 Cos[b0]+a1 Cos[b1]))]]},{x,-2Pi, 3Pi},PlotStyle->{{Black,Thickness[0.01]},{Red,Thickness[0.005]}},PlotRange->{-3,3}],
Row[{
Column[{"a0",Control[{{a0,0.,""},-2,2,ControlType->VerticalSlider}],Dynamic[NumberForm[a0,{2,1},NumberPadding->{" ","0"}]]},Alignment->Center],Column[{"a1",Control[{{a1,1.,""},-2,2,ControlType->VerticalSlider}],Dynamic[NumberForm[a1,{2,1},NumberPadding->{" ","0"}]]},Alignment->Center]
}, Spacer[5]],
Row[{
Column[{"b0",Control[{{b0,0.,""},-Pi,Pi,ControlType->VerticalSlider}],Dynamic[NumberForm[b0,{2,1},NumberPadding->{" ","0"}]]},Alignment->Center],Column[{"b1",Control[{{b1,0.,""},-Pi,Pi,ControlType->VerticalSlider}],Dynamic[NumberForm[b1,{2,1},NumberPadding->{" ","0"}]]},Alignment->Center]
},Spacer[5]],ControlPlacement->{Left,Right} 
]

If you set the parameters a0, b0, a1, b1 such that a0 Cos[b0]+a1 Cos[b1]=0, Mathematica will report errors.