Trigonometry Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \sin (θ) - \cos (θ)$ and $b = \sin (θ) + \cos (θ)$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $\sin (θ) - \cos (θ) + \sin (θ) + \cos (θ)$ .

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Step 2.1.1

Add $- \cos (θ)$ and $\cos (θ)$ .

$(\sin (θ) + 0 + \sin (θ)) (\sin (θ) - \cos (θ) - (\sin (θ) + \cos (θ)))$

Step 2.1.2

Add $\sin (θ)$ and $0$ .

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

Step 2.2

Add $\sin (θ)$ and $\sin (θ)$ .

$2 \sin (θ) (\sin (θ) - \cos (θ) - (\sin (θ) + \cos (θ)))$

Step 2.3

Apply the distributive property.

$2 \sin (θ) (\sin (θ) - \cos (θ) - \sin (θ) - \cos (θ))$

Step 2.4

Combine the opposite terms in $\sin (θ) - \cos (θ) - \sin (θ) - \cos (θ)$ .

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Step 2.4.1

Subtract $\sin (θ)$ from $\sin (θ)$ .

$2 \sin (θ) (0 - \cos (θ) - \cos (θ))$

Step 2.4.2

Subtract $\cos (θ)$ from $0$ .

$2 \sin (θ) (- \cos (θ) - \cos (θ))$

Step 2.5

Subtract $\cos (θ)$ from $- \cos (θ)$ .

$2 \sin (θ) (- 2 \cos (θ))$

Step 2.6

Simplify the expression.

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Step 2.6.1

Reorder $2 \sin (θ)$ and $- 2 \cos (θ)$ .

$- 2 \cos (θ) (2 \sin (θ))$

Step 2.6.2

Add parentheses.

$- 2 (\cos (θ) (2 \sin (θ)))$

Step 2.6.3

Reorder $\cos (θ)$ and $2 \sin (θ)$ .

$- 2 (2 \sin (θ) \cos (θ))$

Step 3

Apply the sine double-angle identity.

$- 2 \sin (2 θ)$