Determine if True sin(x)^4-cos(x)^4=sin(x)^2-cos(x)^2

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

Step 1

Start on the left side.

$\sin^{4} (x) - \cos^{4} (x)$

Step 2

Factor.

Rewrite $\sin^{4} (x)$ as ${(\sin^{2} (x))}^{2}$ .

${(\sin^{2} (x))}^{2} - \cos^{4} (x)$

Rewrite $\cos^{4} (x)$ as ${(\cos^{2} (x))}^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin^{2} (x)$ and $b = \cos^{2} (x)$ .

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

Simplify.

Apply pythagorean identity.

$1 (\sin^{2} (x) - \cos^{2} (x))$

Multiply $\sin^{2} (x) - \cos^{2} (x)$ by $1$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (x)$ and $b = \cos (x)$ .

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

Step 3

Simplify.

Expand $(\sin (x) + \cos (x)) (\sin (x) - \cos (x))$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin^{4} (x) - \cos^{4} (x) = \sin^{2} (x) - \cos^{2} (x)$ is an identity

Determine if True sin(x)^4-cos(x)^4=sin(x)^2-cos(x)^2

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