You are given

cos(A) - sin(A) = sqrt(2)*sin(A).    (1)


Divide both sides by . You will get

 -  = sin(A).


It is the same as

 -  = sin(A).      (2)


Now recall that  =  = .


Therefore, you can re-write (2) in the form

 -  = sin(A).


Using the adding/subtracting formula for sine, it is the same as

 = ,                         (3)


which implies EITHER

     =  + ,                     (4)    

OR

     +  =                    (5)

where k is any integer.


Equation (5) has no solution, obviously.

Equation (4) has the solution

    2A = ,   or   A = .      (6)


Actually, we have two cases:  A =   and  A = .


It is well known fact that 

 = ,   = .

    (see the lesson Miscellaneous Trigonometry problems in this site).


So, if A = ,   then  cos(A) + sin(A) =  + .


    If A = ,  then  cos(A) + sin(A) = -(  +  ).