Precalculus Examples

$\sin (a + b) \sin (a - b)$

Step 1

Apply the sum of angles identity.

$(\sin (a) \cos (b) + \cos (a) \sin (b)) \sin (a - b)$

Step 2

Apply the difference of angles identity.

$(\sin (a) \cos (b) + \cos (a) \sin (b)) (\sin (a) \cos (b) - \cos (a) \sin (b))$

Step 3

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

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Step 3.1

Apply the distributive property.

$\sin (a) \cos (b) (\sin (a) \cos (b) - \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b) - \cos (a) \sin (b))$

Step 3.2

Apply the distributive property.

$\sin (a) \cos (b) (\sin (a) \cos (b)) + \sin (a) \cos (b) (- \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b) - \cos (a) \sin (b))$

Step 3.3

Apply the distributive property.

$\sin (a) \cos (b) (\sin (a) \cos (b)) + \sin (a) \cos (b) (- \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $\sin (a) \cos (b) (\sin (a) \cos (b)) + \sin (a) \cos (b) (- \cos (a) \sin (b)) + \cos (a) \sin (b) (\sin (a) \cos (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$ .

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Step 4.1.1

Reorder the factors in the terms $\sin (a) \cos (b) (- \cos (a) \sin (b))$ and $\cos (a) \sin (b) (\sin (a) \cos (b))$ .

$\sin (a) \cos (b) (\sin (a) \cos (b)) - \cos (a) \cos (b) \sin (a) \sin (b) + \cos (a) \cos (b) \sin (a) \sin (b) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Step 4.1.2

Add $- \cos (a) \cos (b) \sin (a) \sin (b)$ and $\cos (a) \cos (b) \sin (a) \sin (b)$ .

$\sin (a) \cos (b) (\sin (a) \cos (b)) + 0 + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Step 4.1.3

Add $\sin (a) \cos (b) (\sin (a) \cos (b))$ and $0$ .

$\sin (a) \cos (b) (\sin (a) \cos (b)) + \cos (a) \sin (b) (- \cos (a) \sin (b))$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $\sin (a) \cos (b) (\sin (a) \cos (b))$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Add $1$ and $1$ .

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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

Step 4.2.1.7

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

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

Step 4.2.1.8

Add $1$ and $1$ .

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

Step 4.2.2

Rewrite using the commutative property of multiplication.

$\sin^{2} (a) \cos^{2} (b) - \cos (a) \sin (b) \cos (a) \sin (b)$

Step 4.2.3

Multiply $- \cos (a) \sin (b) \cos (a)$ .

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Step 4.2.3.1

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

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

Step 4.2.3.2

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

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

Step 4.2.3.3

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

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

Step 4.2.3.4

Add $1$ and $1$ .

$\sin^{2} (a) \cos^{2} (b) - \cos^{2} (a) \sin (b) \sin (b)$

Step 4.2.4

Multiply $- \cos^{2} (a) \sin (b) \sin (b)$ .

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Step 4.2.4.1

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

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

Step 4.2.4.2

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

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

Step 4.2.4.3

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

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

Step 4.2.4.4

Add $1$ and $1$ .

$\sin^{2} (a) \cos^{2} (b) - \cos^{2} (a) \sin^{2} (b)$