Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Add $- \cos (x)$ and $\cos (x)$ .

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

Step 2.1.2

Add $\sin (x)$ and $0$ .

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

Step 2.2

Add $\sin (x)$ and $\sin (x)$ .

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

Step 2.3

Apply the distributive property.

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Subtract $\cos (x)$ from $0$ .

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

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

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

Step 2.6.2

Add parentheses.

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

Step 2.6.3

Reorder $\cos (x)$ and $2 \sin (x)$ .

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

Step 3

Apply the sine double-angle identity.

$- 2 \sin (2 x)$