Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

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

Step 3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

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

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

Since $\cos (- b)$ is an even function, rewrite $\cos (- b)$ as $\cos (b)$ .

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

Step 4.1.2

Since $\sin (- b)$ is an odd function, rewrite $\sin (- b)$ as $- \sin (b)$ .

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

Step 4.1.3

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

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$(\cos (a) \cos (b) - \sin (a) \sin (b)) \cdot (\cos (a) \cos (b) + 1 \sin (a) \sin (b))$

Step 4.1.3.2

Multiply $\sin (a)$ by $1$ .

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Subtract $\cos (a) \cos (b) \sin (a) \sin (b)$ from $\cos (a) \cos (b) \sin (a) \sin (b)$ .

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

Step 4.3.3

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

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

Step 4.4

Simplify each term.

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Step 4.4.1

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

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Step 4.4.1.1

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

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

Step 4.4.1.2

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

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

Step 4.4.1.3

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

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

Step 4.4.1.4

Add $1$ and $1$ .

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

Step 4.4.1.5

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

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

Step 4.4.1.6

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

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

Step 4.4.1.7

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

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

Step 4.4.1.8

Add $1$ and $1$ .

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

Step 4.4.2

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

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Step 4.4.2.1

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

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

Step 4.4.2.2

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

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

Step 4.4.2.3

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

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

Step 4.4.2.4

Add $1$ and $1$ .

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

Step 4.4.2.5

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

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

Step 4.4.2.6

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

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

Step 4.4.2.7

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

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

Step 4.4.2.8

Add $1$ and $1$ .

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

Apply the distributive property.

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

Step 7

Multiply ${\cos (b)}^{2}$ by $1$ .

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

Step 8

Apply Pythagorean identity.

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Step 8.1

Factor $- \sin^{2} (a)$ out of $- \sin^{2} (a) \cos^{2} (b)$ .

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

Step 8.2

Factor $- \sin^{2} (a)$ out of $- \sin^{2} (a) \sin^{2} (b)$ .

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

Step 8.3

Factor $- \sin^{2} (a)$ out of $- \sin^{2} (a) (\cos^{2} (b)) - \sin^{2} (a) (\sin^{2} (b))$ .

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

Step 8.4

Rearrange terms.

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

Step 8.5

Apply pythagorean identity.

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

Step 9

Multiply $- 1$ by $1$ .

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

Step 10

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

$\cos (a + b) \cdot \cos (a - b) = \cos^{2} (b) - \sin^{2} (a)$ is an identity