How do you factor and simplify ${sin}^{4} x - {cos}^{4} x$ $sin^4x-cos^4x$ ?

Question

How do you factor and simplify ${sin}^{4} x - {cos}^{4} x$ $sin^4x-cos^4x$ ?

2 Answers

Impact of this question

117998 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-29T22:02:44

$(sin x - cos x) (sin x + cos x)$ $(sinx-cosx)(sinx+cosx)$

Explanation:

Factorizing this algebraic expression is based on this property:

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

Taking ${sin}^{2} x = a$ $sin^2x =a$ and ${cos}^{2} x = b$ $cos^2x=b$ we have :

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

Applying the above property we have:

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

Applying the same property on ${sin}^{2} x - {cos}^{2} x$ $sin^2x-cos^2x$

thus,

${({sin}^{2} x)}^{2} - {({cos}^{2} x)}^{2}$ $(sin^2x)^2-(cos^2x)^2$
$= (sin x - cos x) (sin x + cos x) ({sin}^{2} x + {cos}^{2} x)$ $=(sinx-Cosx)(sinx+cosx)(sin^2x+cos^2x)$

Knowing the Pythagorean identity, ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x=1$ we simplify the expression so,

${({sin}^{2} x)}^{2} - {({cos}^{2} x)}^{2}$ $(sin^2x)^2-(cos^2x)^2$
$= (sin x - cos x) (sin x + cos x) ({sin}^{2} x + {cos}^{2} x)$ $=(sinx-Cosx)(sinx+cosx)(sin^2x+cos^2x)$
$= (sin x - cos x) (sin x + cos x) (1)$ $=(sinx-cosx)(sinx+cosx)(1)$
$= (sin x - cos x) (sin x + cos x)$ $=(sinx-cosx)(sinx+cosx)$

Therefore,
${sin}^{4} x - {cos}^{4} x = (sin x - cos x) (sin x + cos x)$ $sin^4x-cos^4x=(sinx-cosx)(sinx+cosx)$

Answer 2 · 2017-04-18T05:34:23

= - cos 2x

Explanation:

${sin}^{4} x - {cos}^{4} x = ({sin}^{2} x + {cos}^{2} x) ({sin}^{2} x - {cos}^{2} x)$ $sin^4x - cos^4 x = (sin^2 x + cos ^2 x)(sin^2 x - cos^2 x)$
Reminder:
${sin}^{2} x + {cos}^{2} x = 1$ $sin^2 x + cos^2 x = 1$ , and
${cos}^{2} x - {sin}^{2} x = cos 2 x$ $cos^2 x - sin^2 x = cos 2x$
Therefore:
${sin}^{4} x - {cos}^{4} x = - cos 2 x$ $sin^4x - cos^4 x = - cos 2x$

Science

Math

Humanities

... and beyond

How do you factor and simplify ${sin}^{4} x - {cos}^{4} x$ $sin^4x-cos^4x$ ?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you factor and simplify sin4x−cos4xsin^4x-cos^4x?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

How do you factor and simplify ${sin}^{4} x - {cos}^{4} x$ $sin^4x-cos^4x$ ?