Find the best lower and upper bounds for

$$\left(\cos A-\sin A\right)\left(\cos B-\sin B\right)\left(\cos C-\sin C\right),$$

$1)~~$ overall acute-angled $\Delta ABC.$

first of all we know that $\cos(A)-\sin(A) = -\sqrt{2}\sin(A - \frac{\pi}{4})$

$\Pi { -\sqrt{2}\sin(A - \frac{\pi}{4})}=-2\sqrt{2}\*\Pi{(\sin(A-\frac{\pi}{4}))}$

$-\frac{1}{2}\*\sqrt{2}<\sin(A-\frac{\pi}{4})<\frac{1}{2}\*\sqrt{2}$

$-\frac{1}{2}\*\sqrt{2}<\sin(B-\frac{\pi}{4})<\frac{1}{2}\*\sqrt{2}$

$-\frac{1}{2}\*\sqrt{2}<\sin(C-\frac{\pi}{4})<\frac{1}{2}\*\sqrt{2}$

Then Lagrange? or there is any other method you all could suggest

Edit:

For Everyone who wanna post or finish using every possible method just write it down here