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Step-by-step explanation:
[tex]\cos^{4} x - \sin^{4} x[/tex]
[tex]= ( \cos^{2} x + \sin^{2} x)( \cos^{2} x - \sin^{2} x)[/tex]
[tex] = (1 - \sin^{2} x + \sin^{2} x)( 1 - \sin^{2} x - \sin^{2} x)[/tex]
[tex] = 1 - 2\sin^{2} x[/tex]
Note: I replaced all cos^2x terms using the identity
[tex]\cos^{2} x = 1 - \sin^{2} x[/tex]