factor $\sin ^2x-\cos ^22x+\cos ^23x-\sin ^24x$

Question

Factor $$\sin ^2x-\cos ^22x+\cos ^23x-\sin ^24x$$

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I got $$\frac{1}{2}\left(\cos 6x+\cos 8x-\cos 2x-\cos 4x\right)$$ but i don't know how to continue

Answer 1

As

$$6+4=8+2$$ use Prosthaphaeresis Formulas on $$\cos6x-\cos4x\text{ and on }\ \ \cos8x-\cos2x$$

Alternatively as $x+4x=2x+3x$

use Prove $ \sin(A+B)\sin(A-B)=\sin^2A-\sin^2B $

$$\sin^2x-\sin^24x=-\sin(4x-x)\sin(4x+x)$$

and $$\cos^22x-\cos^23x=1-\sin^22x-(1-\sin^23x)=\sin(3x+2x)\sin(3x-2x)$$