Help me to find the value of $\frac{\sin^3(\theta)-\cos^3(\theta)}{\sin(\theta-\pi/4)}$ if $\sin(2\theta)=2\sqrt{2}-2$

Question

What is the value of the expression $$\frac{\sin^3(\theta)-\cos^3(\theta)}{\sin(\theta-\pi/4)}$$ if $$\sin(2\theta)=2\sqrt{2}-2$$ I tried some approaches but none seem to work.

Answer 1

$$\frac{\sin^3\theta-\cos^3\theta}{\sin(\theta-\pi/4)}=\sqrt{2}\frac{(\sin\theta-\cos\theta)(\sin^2\theta+\sin\theta\cos\theta+\cos^2\theta)}{\sin\theta-\cos\theta}=\sqrt2(1+\frac12\sin2\theta)$$