Starting from

    sin(x) + cos(x) = 1,      (1)


square both sides

     = 1.      (2)



Take into account that   +  == 1.


You will get then from  (2)


    2*sin(x)*cos(x) = 0,

or

      sin(x(*cos(x) = 0.


So, equation (1) implies

    sin(x) = 0  or  cos(x) = 0.


It gives  x =   or  x = ,   where k is any integer number.


Checking these potential solutions,  we get the final answers  x = ,  or  x = ,  where n is any integer number.


Of the given four optional choices,  only  x= 90°  is the solutions to the given equation.