Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(\cos (x) + \cos (y))}^{2}$ as $(\cos (x) + \cos (y)) (\cos (x) + \cos (y))$ .

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

Add $1$ and $1$ .

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

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

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

Step 1.3.1.2.2

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

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

Step 1.3.1.2.3

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

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

Step 1.3.1.2.4

Add $1$ and $1$ .

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

Step 1.3.2

Reorder the factors of $\cos (y) \cos (x)$ .

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

Step 1.3.3

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

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

Step 1.4

Rewrite ${(\sin (x) - \sin (y))}^{2}$ as $(\sin (x) - \sin (y)) (\sin (x) - \sin (y))$ .

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

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

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Step 1.6.1.1.1

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

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

Step 1.6.1.1.2

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

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

Step 1.6.1.1.3

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

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

Step 1.6.1.1.4

Add $1$ and $1$ .

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

Step 1.6.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.3

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

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Step 1.6.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.3.2

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

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

Step 1.6.1.3.3

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

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

Step 1.6.1.3.4

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

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

Step 1.6.1.3.5

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

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

Step 1.6.1.3.6

Add $1$ and $1$ .

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

Step 1.6.2

Reorder the factors of $- \sin (y) \sin (x)$ .

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

Step 1.6.3

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

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

Step 2

Move $\sin^{2} (x)$ .

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

Step 3

Rearrange terms.

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

Step 4

Apply pythagorean identity.

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

Step 5

Move $\sin^{2} (y)$ .

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

Step 6

Rearrange terms.

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

Step 7

Apply pythagorean identity.

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

Step 8

Add $1$ and $1$ .

$2 \cos (x) \cos (y) + 2 - 2 \sin (x) \sin (y)$