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

Then teach the underlying concepts
Don't copy without citing sources
preview
?

Explanation

Explain in detail...

Explanation:

I want someone to double check my answer

34
Nghi N Share
Apr 18, 2017

= - cos 2x

Explanation:

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

Then teach the underlying concepts
Don't copy without citing sources
preview
?

Explanation

Explain in detail...

Explanation:

I want someone to double check my answer

24
Apr 18, 2017

$\left(\sin x - \cos x\right) \left(\sin x + \cos x\right)$

Explanation:

Factorizing this algebraic expression is based on this property:

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

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

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

Applying the above property we have:

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

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

thus,

${\left({\sin}^{2} x\right)}^{2} - {\left({\cos}^{2} x\right)}^{2}$
$= \left(\sin x - C o s x\right) \left(\sin x + \cos x\right) \left({\sin}^{2} x + {\cos}^{2} x\right)$

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

${\left({\sin}^{2} x\right)}^{2} - {\left({\cos}^{2} x\right)}^{2}$
$= \left(\sin x - C o s x\right) \left(\sin x + \cos x\right) \left({\sin}^{2} x + {\cos}^{2} x\right)$
$= \left(\sin x - \cos x\right) \left(\sin x + \cos x\right) \left(1\right)$
$= \left(\sin x - \cos x\right) \left(\sin x + \cos x\right)$

Therefore,
${\sin}^{4} x - {\cos}^{4} x = \left(\sin x - \cos x\right) \left(\sin x + \cos x\right)$

• 20 minutes ago
• 21 minutes ago
• 27 minutes ago
• 33 minutes ago
• 3 minutes ago
• 5 minutes ago
• 11 minutes ago
• 13 minutes ago
• 19 minutes ago
• 19 minutes ago
• 20 minutes ago
• 21 minutes ago
• 27 minutes ago
• 33 minutes ago