Question

Prove tan(x - (π / 4)) = (sin x – cos x) / (cos x + sin x) by filling in the reasons in the table below.

Answer 1

No way to know what reasons you're supposed to choose from...

By definition of tangent,

$\tan\left(x-\dfrac\pi4\right)=\dfrac{\sin\left(x-\frac\pi4\right)}{\cos\left(x-\frac\pi4\right)}$

The angle sum identities give

$\tan\left(x-\dfrac\pi4\right)=\dfrac{\sin x\cos\frac\pi4-\cos x\sin\frac\pi4}{\cos x\cos\frac\pi4+\sin x\sin\frac\pi4}$

$cos\dfrac\pi4=\sin\dfrac\pi4=\dfrac1{\sqrt2}$, so we can cancel those terms to get

$\tan\left(x-\dfrac\pi4\right)=\dfrac{\sin x-\cos x}{\sin x+\cos x}$

as required.