Question

$\sqrt{2}\cos A=\cos B+{\cos }^{3}B$
$\sqrt{2}\sin A=\sin B-{\sin }^{3}B$
Find $|\sin (A-B)|$.

Answer 1

$\cos A=\frac{1}{\sqrt{2}}(\cos B+{\cos }^{3}B)$

$\sin A=\frac{1}{\sqrt{2}}(\sin B-{\sin }^{3}B)$

$|\sin (A-B)|=|\sin A\cos B-\cos A\sin B|=\frac{1}{\sqrt{2}}|\sin B\cos B-{\sin }^{3}B\cos B-\sin B\cos B-\sin B{\cos }^{3}B|$

$=\frac{1}{\sqrt{2}}|\sin B\cos B|=\frac{1}{2\sqrt{2}}|\sin 2B|$