Trigonometry Examples

${(\cos (θ) - \sin (θ))}^{2}$

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $\cos (θ) \cos (θ)$ .

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

Add $1$ and $1$ .

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

Step 3.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.3

Multiply $- \sin (θ) (- \sin (θ))$ .

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Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

Multiply $\sin (θ)$ by $1$ .

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

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

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

Step 3.1.3.6

Add $1$ and $1$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Move $\sin^{2} (θ)$ .

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

Step 5

Rearrange terms.

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

Step 6

Apply pythagorean identity.

$1 - 2 \cos (θ) \sin (θ)$