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-\frac\pi4\right)=(\sin\left(x-\frac\pi4\right))/(\cos\left(x-\frac\pi4\right))`

The angle sum identities give

`\tan\left(x-\frac\pi4\right)=(\sin x\cos\frac\pi4-\cos x\sin\frac\pi4)/(\cos x\cos\frac\pi4+\sin x\sin\frac\pi4)`

`cos\frac\pi4=\sin\frac\pi4=\frac1{\sqrt2}`, so we can cancel those terms to get

`\tan\left(x-\frac\pi4\right)=(\sin x-\cos x)/(\sin x+\cos x)`

as required.