Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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

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Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

Add $1$ and $1$ .

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

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.3

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

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Step 1.3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.3.2

Multiply $\cos (x)$ by $1$ .

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

Step 1.3.1.3.3

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

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

Step 1.3.1.3.4

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

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

Step 1.3.1.3.5

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

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

Step 1.3.1.3.6

Add $1$ and $1$ .

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

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

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

Step 1.5

Apply pythagorean identity.

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Apply the distributive property.

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

Step 1.7.3

Apply the distributive property.

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

Step 1.8

Simplify and combine like terms.

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

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

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Step 1.8.1.1.1

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

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

Step 1.8.1.1.2

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

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

Step 1.8.1.1.3

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

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

Step 1.8.1.1.4

Add $1$ and $1$ .

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

Step 1.8.1.2

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

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Step 1.8.1.2.1

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

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

Step 1.8.1.2.2

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

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

Step 1.8.1.2.3

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

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

Step 1.8.1.2.4

Add $1$ and $1$ .

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

Step 1.8.2

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

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

Step 1.8.3

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

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

Step 1.9

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

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

Step 1.10

Apply pythagorean identity.

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

Step 1.11

Simplify each term.

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Step 1.11.1

Reorder $2 \cos (x)$ and $\sin (x)$ .

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

Step 1.11.2

Reorder $\sin (x)$ and $2$ .

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

Step 1.11.3

Apply the sine double-angle identity.

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

Step 2

Add $1$ and $1$ .

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