Precalculus Examples

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

Step 1

Rewrite ${(\sin (θ) + \cos (θ))}^{2}$ as $(\sin (θ) + \cos (θ)) (\sin (θ) + \cos (θ))$ .

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

Step 2

Expand $(\sin (θ) + \cos (θ)) (\sin (θ) + \cos (θ))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\sin (θ) \sin (θ)$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

Multiply $\cos (θ) \cos (θ)$ .

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

Add $1$ and $1$ .

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

Step 3.2

Reorder the factors of $\sin (θ) \cos (θ)$ .

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

Step 3.3

Add $\cos (θ) \sin (θ)$ and $\cos (θ) \sin (θ)$ .

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

Step 4

Move $\cos^{2} (θ)$ .

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

Step 5

Apply pythagorean identity.

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

Step 6

Simplify each term.

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Step 6.1

Reorder $2 \cos (θ)$ and $\sin (θ)$ .

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

Step 6.2

Remove parentheses.

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

Step 6.3

Reorder $\sin (θ)$ and $2$ .

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

Step 6.4

Apply the sine double-angle identity.

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

Step 7

Rewrite ${(\sin (θ) - \cos (θ))}^{2}$ as $(\sin (θ) - \cos (θ)) (\sin (θ) - \cos (θ))$ .

$1 + \sin (2 θ) + (\sin (θ) - \cos (θ)) (\sin (θ) - \cos (θ))$

Step 8

Expand $(\sin (θ) - \cos (θ)) (\sin (θ) - \cos (θ))$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$1 + \sin (2 θ) + \sin (θ) (\sin (θ) - \cos (θ)) - \cos (θ) (\sin (θ) - \cos (θ))$

Step 8.2

Apply the distributive property.

$1 + \sin (2 θ) + \sin (θ) \sin (θ) + \sin (θ) (- \cos (θ)) - \cos (θ) (\sin (θ) - \cos (θ))$

Step 8.3

Apply the distributive property.

$1 + \sin (2 θ) + \sin (θ) \sin (θ) + \sin (θ) (- \cos (θ)) - \cos (θ) \sin (θ) - \cos (θ) (- \cos (θ))$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $\sin (θ) \sin (θ)$ .

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Step 9.1.1.1

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

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

Step 9.1.1.2

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

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

Step 9.1.1.3

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

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

Step 9.1.1.4

Add $1$ and $1$ .

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

Step 9.1.2

Rewrite using the commutative property of multiplication.

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

Step 9.1.3

Multiply $- \cos (θ) (- \cos (θ))$ .

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Step 9.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 9.1.3.2

Multiply $\cos (θ)$ by $1$ .

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

Step 9.1.3.3

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

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

Step 9.1.3.4

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

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

Step 9.1.3.5

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

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

Step 9.1.3.6

Add $1$ and $1$ .

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

Step 9.2

Reorder the factors of $- \sin (θ) \cos (θ)$ .

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

Step 9.3

Subtract $\cos (θ) \sin (θ)$ from $- \cos (θ) \sin (θ)$ .

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

Step 10

Move $\cos^{2} (θ)$ .

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

Step 11

Apply pythagorean identity.

$1 + \sin (2 θ) + 1 - 2 \cos (θ) \sin (θ)$

Step 12

Add $1$ and $1$ .

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