Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $1$ .

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

Add $1$ and $1$ .

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

Step 3.2

Subtract $\cos (t)$ from $\cos (t)$ .

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

Step 3.3

Add $1$ and $0$ .

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

Step 4

Apply pythagorean identity.

$\sin^{2} (t)$