You want the value of x for which sin x - cos x = 0. Why are you taking the cases where cos x = 1, sin x = 0, etc.
Instead, solve for x.
sin x - cos x = 0
=> sin x = cos x
=> sin x / cos x = 1
=> tan x = 1
=> x = arc tan 1
=> x = pi/4 + n*pi
So the solution of the equation is x = pi/4 + n*pi
Your tutor is correct! There are not just 2 possibilities for the angle x.
It is obvious that the values for sin x and cos x have to be equal in order to cancel the expression sin x - cos x.
sin x - cos x=0
We could divide the equation by (cos x) and the result will be:
sin x /cos x - 1 =0
But we know that the ratio between sinx and cos x determine the tangent function.
tan x -1 =0
tan x = 1
x= arctan 1 + k*pi
x = pi/4 + k*pi
We could also notice that if the ratio sin x / cos x=1, it means that the numerator and denominator are equal.
=> sin x = cosx!
If sin x = cos x => the x values are equal, too, so, in a right angle triangle, the angles could only be of 45 degrees (the conclusion is based on fact that in a triangle, the sum of angles is 180 degrees, and one of them is 90 degrees and the other 2 are equals, so
90 + 2*x=180