sinx- sqrt(2)cosx= 1 is of the form acosx+ bsinx= c. Comparing the two equations we have c=1. But by a^2+ b^2= c^2 we have c= sqrt(3). If c=1 and a^2+ b^2= c^2, then 1^2+ (-sqrt(2))^2= sqrt(3) ≠ 1.
cosx= x/r where r=1, so cosx=x and sinx= y. So I thought that must mean acosx=x and bsinx= y...