Trig3.1

Prove that $\frac{\cos 3A+\sin 3A}{\cos A-\sin A}=1+2\sin 2A\,$

LHS= $\frac{4\cos^3 A-3\cos A+3\sin A-4\sin^3 A}{\cos A-\sin A}\,$

$\frac{4(\cos^3 A-\sin^3 A)-3(\cos A-\sin A)}{\cos A-\sin A}\,$

$\frac{(\cos A-\sin A)(4(\cos^2 A+\cos A \sin A+\sin^2 A)-3}{\cos A-\sin A}\,$

$4(1+\sin A \cos A)-3\,$

$4+4\sin A \cos A-3\,$

$1+2\sin 2A\,$ =RHS