No way to know what reasons you're supposed to choose from...
By definition of tangent,
[tex]\tan\left(x-\dfrac\pi4\right)=\dfrac{\sin\left(x-\frac\pi4\right)}{\cos\left(x-\frac\pi4\right)}[/tex]
The angle sum identities give
[tex]\tan\left(x-\dfrac\pi4\right)=\dfrac{\sin x\cos\frac\pi4-\cos x\sin\frac\pi4}{\cos x\cos\frac\pi4+\sin x\sin\frac\pi4}[/tex]
[tex]cos\dfrac\pi4=\sin\dfrac\pi4=\dfrac1{\sqrt2}[/tex], so we can cancel those terms to get
[tex]\tan\left(x-\dfrac\pi4\right)=\dfrac{\sin x-\cos x}{\sin x+\cos x}[/tex]
as required.
