How to factor and simplify sin4x−cos4x?

Question

Gremoli9bs · Accepted Answer

This algebraic expression can be factored based on the following property:

a^{2} - b^{2} = (a - b) (a + b)

Taking

\sin^{2} x = a

and

\cos^{2} x = b

we have :

\sin^{4} x - \cos^{4} x = {(\sin^{2} x)}^{2} - {(\cos^{2} x)}^{2} = a^{2} - b^{2}

Applying the above property we have:

{(\sin^{2} x)}^{2} - {(\cos^{2} x)}^{2} = (\sin^{2} x - \cos^{2} x) (\sin^{2} x + \cos^{2} x)

Applying the same property on

\sin^{2} x - \cos^{2} x

thus,

{(\sin^{2} x)}^{2} - {(\cos^{2} x)}^{2}

= (\sin x - \cos x) (\sin x + \cos x) (\sin^{2} x + \cos^{2} x)

Knowing the Pythagorean identity,

\sin^{2} x + \cos^{2} x = 1

we simplify the expression so,

{(\sin^{2} x)}^{2} - {(\cos^{2} x)}^{2}

= (\sin x - \cos x) (\sin x + \cos x) (\sin^{2} x + \cos^{2} x)

= (\sin x - \cos x) (\sin x + \cos x) (1)

= (\sin x - \cos x) (\sin x + \cos x)

Hence,

\sin^{4} x - \cos^{4} x = (\sin x - \cos x) (\sin x + \cos x)

King Waters · Accepted Answer

sin4x−cos4x=(sin2x+cos2x)(sin2x−cos2x)Reminder:sin2x+cos2x=1, andcos2x−sin2x=cos⁡2xTherefore:sin4x−cos4x=−cos⁡2x

How to factor and simplify sin^4x-cos^4x?