Precalculus Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

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

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Apply the distributive property.

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

Step 2.1.3

Apply the distributive property.

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

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $1$ by $1$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.2.1.5

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

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Step 2.2.1.5.1

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

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

Step 2.2.1.5.2

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

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

Step 2.2.1.5.3

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

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

Step 2.2.1.5.4

Add $1$ and $1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Add $1$ and $0$ .

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

Step 2.3

Apply pythagorean identity.

$\cos^{2} (a)$

Step 3

Because the two sides have been shown to be equivalent, the equation is an identity.

$(1 + \sin (a)) (1 - \sin (a)) = \cos^{2} (a)$ is an identity