Simplify (1-sin(theta))(1+sin(theta))

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Trigonometry

$(1 - \sin (θ)) (1 + \sin (θ))$

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

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

Multiply $\sin (θ)$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $- \sin (θ) \sin (θ)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Subtract $\sin (θ)$ from $\sin (θ)$ .

$1 + 0 - \sin^{2} (θ)$

Add $1$ and $0$ .

$1 - \sin^{2} (θ)$

Apply pythagorean identity.

$\cos^{2} (θ)$

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