Precalculus Examples

$(1 + \sin (y)) (1 + \sin (- y))$

Step 1

Since $\sin (- y)$ is an odd function, rewrite $\sin (- y)$ as $- \sin (y)$ .

$(1 + \sin (y)) (1 - \sin (y))$

Step 2

Expand $(1 + \sin (y)) (1 - \sin (y))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$1 (1 - \sin (y)) + \sin (y) (1 - \sin (y))$

Step 2.2

Apply the distributive property.

$1 \cdot 1 + 1 (- \sin (y)) + \sin (y) (1 - \sin (y))$

Step 2.3

Apply the distributive property.

$1 \cdot 1 + 1 (- \sin (y)) + \sin (y) \cdot 1 + \sin (y) (- \sin (y))$

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 $1$ .

$1 + 1 (- \sin (y)) + \sin (y) \cdot 1 + \sin (y) (- \sin (y))$

Step 3.1.2

Multiply $- \sin (y)$ by $1$ .

$1 - \sin (y) + \sin (y) \cdot 1 + \sin (y) (- \sin (y))$

Step 3.1.3

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

$1 - \sin (y) + \sin (y) + \sin (y) (- \sin (y))$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$1 - \sin (y) + \sin (y) - \sin (y) \sin (y)$

Step 3.1.5

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

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Step 3.1.5.1

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

$1 - \sin (y) + \sin (y) - (\sin^{1} (y) \sin (y))$

Step 3.1.5.2

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

$1 - \sin (y) + \sin (y) - (\sin^{1} (y) \sin^{1} (y))$

Step 3.1.5.3

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

$1 - \sin (y) + \sin (y) - {\sin (y)}^{1 + 1}$

Step 3.1.5.4

Add $1$ and $1$ .

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

Step 3.2

Add $- \sin (y)$ and $\sin (y)$ .

$1 + 0 - \sin^{2} (y)$

Step 3.3

Add $1$ and $0$ .

$1 - \sin^{2} (y)$

Step 4

Apply pythagorean identity.

$\cos^{2} (y)$