Precalculus Examples

$\cos^{4} (θ) - \sin^{4} (θ) = 1 - 2 \sin^{2} (θ)$

Step 1

Start on the left side.

$\cos^{4} (θ) - \sin^{4} (θ)$

Step 2

Factor.

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Step 2.1

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

${(\cos^{2} (θ))}^{2} - \sin^{4} (θ)$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

Rearrange terms.

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

Step 2.4.2

Apply pythagorean identity.

$1 (\cos^{2} (θ) - \sin^{2} (θ))$

Step 2.4.3

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

$\cos^{2} (θ) - \sin^{2} (θ)$

Step 2.4.4

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

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

Step 3

Apply the distributive property.

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Multiply $\cos (θ) \cos (θ)$ .

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Step 4.1.2.1

Raise $\cos (θ)$ to the power of $1$ .

${\cos (θ)}^{1} \cos (θ) + \sin (θ) \cos (θ) + (\cos (θ) + \sin (θ)) (- \sin (θ))$

Step 4.1.2.2

Raise $\cos (θ)$ to the power of $1$ .

${\cos (θ)}^{1} {\cos (θ)}^{1} + \sin (θ) \cos (θ) + (\cos (θ) + \sin (θ)) (- \sin (θ))$

Step 4.1.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\cos (θ)}^{1 + 1} + \sin (θ) \cos (θ) + (\cos (θ) + \sin (θ)) (- \sin (θ))$

Step 4.1.2.4

Add $1$ and $1$ .

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Multiply $\sin (θ) (- \sin (θ))$ .

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Step 4.1.4.1

Raise $\sin (θ)$ to the power of $1$ .

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

Step 4.1.4.2

Raise $\sin (θ)$ to the power of $1$ .

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

Step 4.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.4.4

Add $1$ and $1$ .

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

Step 4.2

Add $\cos (θ) \sin (θ)$ and $\cos (θ) (- \sin (θ))$ .

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Step 4.2.1

Reorder $\cos (θ)$ and $- 1$ .

${\cos (θ)}^{2} + \cos (θ) \sin (θ) - 1 \cdot \cos (θ) \sin (θ) - {\sin (θ)}^{2}$

Step 4.2.2

Subtract $\cos (θ) \sin (θ)$ from $\cos (θ) \sin (θ)$ .

${\cos (θ)}^{2} + 0 - {\sin (θ)}^{2}$

Step 4.3

Add ${\cos (θ)}^{2}$ and $0$ .

$\cos^{2} (θ) - \sin^{2} (θ)$

Step 5

Apply Pythagorean identity in reverse.

$1 - \sin^{2} (θ) - \sin^{2} (θ)$

Step 6

Subtract ${\sin (θ)}^{2}$ from $- {\sin (θ)}^{2}$ .

$1 - 2 \sin^{2} (θ)$

Step 7

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

$\cos^{4} (θ) - \sin^{4} (θ) = 1 - 2 \sin^{2} (θ)$ is an identity