Trigonometry Examples

$(1 + \cos (b)) (1 - \cos (- b))$

Step 1

Since $\cos (- b)$ is an even function, rewrite $\cos (- b)$ as $\cos (b)$ .

$(1 + \cos (b)) (1 - \cos (b))$

Step 2

Expand $(1 + \cos (b)) (1 - \cos (b))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$1 (1 - \cos (b)) + \cos (b) (1 - \cos (b))$

Step 2.2

Apply the distributive property.

$1 \cdot 1 + 1 (- \cos (b)) + \cos (b) (1 - \cos (b))$

Step 2.3

Apply the distributive property.

$1 \cdot 1 + 1 (- \cos (b)) + \cos (b) \cdot 1 + \cos (b) (- \cos (b))$

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 $1$ by $1$ .

$1 + 1 (- \cos (b)) + \cos (b) \cdot 1 + \cos (b) (- \cos (b))$

Step 3.1.2

Multiply $- \cos (b)$ by $1$ .

$1 - \cos (b) + \cos (b) \cdot 1 + \cos (b) (- \cos (b))$

Step 3.1.3

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

$1 - \cos (b) + \cos (b) + \cos (b) (- \cos (b))$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$1 - \cos (b) + \cos (b) - \cos (b) \cos (b)$

Step 3.1.5

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

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Step 3.1.5.1

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

$1 - \cos (b) + \cos (b) - (\cos^{1} (b) \cos (b))$

Step 3.1.5.2

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

$1 - \cos (b) + \cos (b) - (\cos^{1} (b) \cos^{1} (b))$

Step 3.1.5.3

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

$1 - \cos (b) + \cos (b) - {\cos (b)}^{1 + 1}$

Step 3.1.5.4

Add $1$ and $1$ .

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

Step 3.2

Add $- \cos (b)$ and $\cos (b)$ .

$1 + 0 - \cos^{2} (b)$

Step 3.3

Add $1$ and $0$ .

$1 - \cos^{2} (b)$

Step 4

Apply pythagorean identity.

$\sin^{2} (b)$