Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Rewrite ${(\sin (x) - \cos (x))}^{2}$ as $(\sin (x) - \cos (x)) (\sin (x) - \cos (x))$ .

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

Step 2.2

Expand $(\sin (x) - \cos (x)) (\sin (x) - \cos (x))$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

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

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

Step 2.3.1.1.2

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

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

Step 2.3.1.1.3

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

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

Step 2.3.1.1.4

Add $1$ and $1$ .

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

Step 2.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.3

Multiply $- \cos (x) (- \cos (x))$ .

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Step 2.3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.3.2

Multiply $\cos (x)$ by $1$ .

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

Step 2.3.1.3.3

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

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

Step 2.3.1.3.4

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

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

Step 2.3.1.3.5

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

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

Step 2.3.1.3.6

Add $1$ and $1$ .

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

Step 2.3.2

Reorder the factors of $- \sin (x) \cos (x)$ .

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

Step 2.3.3

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

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

Step 2.4

Move $\cos^{2} (x)$ .

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

Step 2.5

Apply pythagorean identity.

$1 - 2 \cos (x) \sin (x)$

Step 3

Rewrite $1 - 2 \cos (x) \sin (x)$ as $1 - 2 \sin (x) \cos (x)$ .

$1 - 2 \sin (x) \cos (x)$

Step 4

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

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