Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $\cos (x) \cos (y) (\cos (x) \cos (y)) + \cos (x) \cos (y) (\sin (x) \sin (y)) - \sin (x) \sin (y) (\cos (x) \cos (y)) - \sin (x) \sin (y) (\sin (x) \sin (y))$ .

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Step 4.1.1

Reorder the factors in the terms $\cos (x) \cos (y) (\sin (x) \sin (y))$ and $- \sin (x) \sin (y) (\cos (x) \cos (y))$ .

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

Step 4.1.2

Subtract $\cos (x) \cos (y) \sin (x) \sin (y)$ from $\cos (x) \cos (y) \sin (x) \sin (y)$ .

$\cos (x) \cos (y) (\cos (x) \cos (y)) + 0 - \sin (x) \sin (y) (\sin (x) \sin (y))$

Step 4.1.3

Add $\cos (x) \cos (y) (\cos (x) \cos (y))$ and $0$ .

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

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $\cos (x) \cos (y) (\cos (x) \cos (y))$ .

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Step 4.2.1.1

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Add $1$ and $1$ .

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

Step 4.2.1.5

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

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

Step 4.2.1.6

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

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

Step 4.2.1.7

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

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

Step 4.2.1.8

Add $1$ and $1$ .

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

Step 4.2.2

Multiply $- \sin (x) \sin (y) (\sin (x) \sin (y))$ .

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

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

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

Step 4.2.2.4

Add $1$ and $1$ .

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

Step 4.2.2.5

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

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

Step 4.2.2.6

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

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

Step 4.2.2.7

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

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

Step 4.2.2.8

Add $1$ and $1$ .

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