Verify if the equations sinx+cosx=0 and sinx-cosx=0 have the same solutions in the set [0,2pi)

Both equations are homogenous equations and they may be
solved using tangent function.

We'll start with the 1st
equation:

sinx+cosx=0

We'll
divide by cos x:

sin x*/cos x + 1 =
0

tan x + 1 = 0

tan x =
-1

x = arctan (-1)

The value
of the tangent function is negative in the 2nd and the 4th
quadrants.

x = - arctan 1

x =
pi - pi/4

x = 3pi/4 (2nd
quadrant)

x = 2pi - pi/4

x =
7pi/4 (4th quadrant)

Now, we'll solve the equation
sinx-cosx=0.

tan x - 1 = 0

tan
x = 1

The value of the tangent function is positive in the
1st and the 3rd quadrants.

x = pi/4 (1st
quadrant)

x = pi + pi/4

x =
5pi/4 (3rd quadrant)

As we can notice, the x
values, that represents the solutions of the given equations, are not the same for both:
the 1st equation allows the set {3pi/4 ; 7pi/4} and the 2nd equation allows the set
{pi/4 ; 5pi/4}.