Precalculus Examples

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

Step 1

Start on the right side.

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

Step 2

Simplify the expression.

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Step 2.1

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

$(\cos (2 u) + \sin (2 u)) (\cos (2 u) - \sin (2 u))$

Step 2.2

Expand $(\cos (2 u) + \sin (2 u)) (\cos (2 u) - \sin (2 u))$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\cos (2 u) (\cos (2 u) - \sin (2 u)) + \sin (2 u) (\cos (2 u) - \sin (2 u))$

Step 2.2.2

Apply the distributive property.

$\cos (2 u) \cos (2 u) + \cos (2 u) (- \sin (2 u)) + \sin (2 u) (\cos (2 u) - \sin (2 u))$

Step 2.2.3

Apply the distributive property.

$\cos (2 u) \cos (2 u) + \cos (2 u) (- \sin (2 u)) + \sin (2 u) \cos (2 u) + \sin (2 u) (- \sin (2 u))$

Step 2.3

Combine the opposite terms in $\cos (2 u) \cos (2 u) + \cos (2 u) (- \sin (2 u)) + \sin (2 u) \cos (2 u) + \sin (2 u) (- \sin (2 u))$ .

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Step 2.3.1

Reorder the factors in the terms $\cos (2 u) (- \sin (2 u))$ and $\sin (2 u) \cos (2 u)$ .

$\cos (2 u) \cos (2 u) - \cos (2 u) \sin (2 u) + \cos (2 u) \sin (2 u) + \sin (2 u) (- \sin (2 u))$

Step 2.3.2

Add $- \cos (2 u) \sin (2 u)$ and $\cos (2 u) \sin (2 u)$ .

$\cos (2 u) \cos (2 u) + 0 + \sin (2 u) (- \sin (2 u))$

Step 2.3.3

Add $\cos (2 u) \cos (2 u)$ and $0$ .

$\cos (2 u) \cos (2 u) + \sin (2 u) (- \sin (2 u))$

Step 2.4

Simplify each term.

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Step 2.4.1

Multiply $\cos (2 u) \cos (2 u)$ .

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Step 2.4.1.1

Raise $\cos (2 u)$ to the power of $1$ .

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

Step 2.4.1.2

Raise $\cos (2 u)$ to the power of $1$ .

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

Step 2.4.1.3

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

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

Step 2.4.1.4

Add $1$ and $1$ .

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

Step 2.4.2

Rewrite using the commutative property of multiplication.

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

Step 2.4.3

Multiply $- \sin (2 u) \sin (2 u)$ .

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Step 2.4.3.1

Raise $\sin (2 u)$ to the power of $1$ .

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

Step 2.4.3.2

Raise $\sin (2 u)$ to the power of $1$ .

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

Step 2.4.3.3

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

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

Step 2.4.3.4

Add $1$ and $1$ .

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

Step 2.5

Apply the cosine double-angle identity.

$\cos (2 (2 u))$

Step 2.6

Multiply $2$ by $2$ .

$\cos (4 u)$

Step 3

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

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