Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

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

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

Add $1$ and $1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

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

Step 5

Rearrange terms.

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

Step 6

Apply pythagorean identity.

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