Trigonometry Examples

$5 (1 - \sin (x)) (1 + \sin (x))$

Step 1

Simplify by multiplying through.

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Step 1.1

Apply the distributive property.

$(5 \cdot 1 + 5 (- \sin (x))) (1 + \sin (x))$

Step 1.2

Multiply.

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Step 1.2.1

Multiply $5$ by $1$ .

$(5 + 5 (- \sin (x))) (1 + \sin (x))$

Step 1.2.2

Multiply $- 1$ by $5$ .

$(5 - 5 \sin (x)) (1 + \sin (x))$

Step 2

Expand $(5 - 5 \sin (x)) (1 + \sin (x))$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$5 (1 + \sin (x)) - 5 \sin (x) (1 + \sin (x))$

Step 2.2

Apply the distributive property.

$5 \cdot 1 + 5 \sin (x) - 5 \sin (x) (1 + \sin (x))$

Step 2.3

Apply the distributive property.

$5 \cdot 1 + 5 \sin (x) - 5 \sin (x) \cdot 1 - 5 \sin (x) \sin (x)$

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 $5$ by $1$ .

$5 + 5 \sin (x) - 5 \sin (x) \cdot 1 - 5 \sin (x) \sin (x)$

Step 3.1.2

Multiply $- 5$ by $1$ .

$5 + 5 \sin (x) - 5 \sin (x) - 5 \sin (x) \sin (x)$

Step 3.1.3

Multiply $- 5 \sin (x) \sin (x)$ .

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Step 3.1.3.1

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

$5 + 5 \sin (x) - 5 \sin (x) - 5 (\sin^{1} (x) \sin (x))$

Step 3.1.3.2

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

$5 + 5 \sin (x) - 5 \sin (x) - 5 (\sin^{1} (x) \sin^{1} (x))$

Step 3.1.3.3

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

$5 + 5 \sin (x) - 5 \sin (x) - 5 {\sin (x)}^{1 + 1}$

Step 3.1.3.4

Add $1$ and $1$ .

$5 + 5 \sin (x) - 5 \sin (x) - 5 \sin^{2} (x)$

Step 3.2

Subtract $5 \sin (x)$ from $5 \sin (x)$ .

$5 + 0 - 5 \sin^{2} (x)$

Step 3.3

Add $5$ and $0$ .

$5 - 5 \sin^{2} (x)$

Step 4

Simplify with factoring out.

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Step 4.1

Factor $5$ out of $5$ .

$5 (1) - 5 \sin^{2} (x)$

Step 4.2

Factor $5$ out of $- 5 \sin^{2} (x)$ .

$5 (1) + 5 (- \sin^{2} (x))$

Step 4.3

Factor $5$ out of $5 (1) + 5 (- \sin^{2} (x))$ .

$5 (1 - \sin^{2} (x))$

Step 5

Apply pythagorean identity.

$5 \cos^{2} (x)$