Find $\sin 4x$ if $\cos x$ - $\sin x$ = -$\frac{\sqrt 2}{4\sin x}$

Question

If $\cos x - \sin x = -\frac{\sqrt 2}{4\sin x}$ then find $\sin4x$

Answer 1

Multiplying two sides of the equation to $2 \sin x$ and add $1$ to both sides;

\begin{eqnarray*} \underbrace{2 \cos(x) \sin(x)}_{ \sin(2x)} + \underbrace{1- 2 \sin^2(x)}_{\cos(2x) } = 1- \frac{1}{\sqrt{2}}. \end{eqnarray*}

Now, we should square both sides of the equation then $\sin 4x=\color{blue}{ \frac{1}{2} - \sqrt 2}$