Some cos

$\begin{aligned} \sin{x} + \cos{x} & = 1 \\ (\sin{x} + \cos{x})^2 & = 1^2 \\ \sin^2{x} + 2\sin{x} \cos{x} + \cos^2{x} & = 1 \\ 1+ 2\sin{x} \cos{x} & = 1 \\ \Rightarrow 2\sin{x} \cos{x} & = 0 \end{aligned}$

Now, we have:

$\begin{aligned} |\sin{x} - \cos{x}| & = \left|\sqrt{(\sin{x} - \cos{x})^2}\right| \\ & = \left|\sqrt{\sin^2{x} - 2\sin{x} \cos{x} + \cos^2{x}} \right| \\ & = \left|\sqrt{\sin^2{x} - 0 + \cos^2{x}} \right| \\ & = \left|\sqrt{1} \right| \\ & = \boxed{1} \end{aligned}$

sinx+cosx=1.

Square it on both sides.

sin^2(x)+cos^2(x)+2sinxcosx = 1.

1+2sinxcosx=1.

2sinxcosx=0.

sinx=0 or cosx=0.

In both cases the other becomes 1 or -1.

So, mod(1)=mod(-1)=1.

mod is modulus function whose range is positive real numbers(including zero).