Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

Step 3

Apply the difference of angles identity.

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

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Add $1$ and $1$ .

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.4.2

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

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

Step 4.1.4.3

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

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

Step 4.1.4.4

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

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

Step 4.1.4.5

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

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

Step 4.1.4.6

Add $1$ and $1$ .

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

Step 4.2

Combine the opposite terms in $\sin (x) \cos (x) \cos (y) + \sin^{2} (x) \sin (y) - \cos (x) \sin (x) \cos (y) + \cos^{2} (x) \sin (y)$ .

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Step 4.2.1

Reorder the factors in the terms $\sin (x) \cos (x) \cos (y)$ and $- \cos (x) \sin (x) \cos (y)$ .

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

Step 4.2.2

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

$\sin^{2} (x) \sin (y) + 0 + \cos^{2} (x) \sin (y)$

Step 4.2.3

Add $\sin^{2} (x) \sin (y)$ and $0$ .

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

Step 4.3

Factor $\sin (y)$ out of $\sin^{2} (x) \sin (y) + \cos^{2} (x) \sin (y)$ .

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Step 4.3.1

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

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

Step 4.3.2

Factor $\sin (y)$ out of $\cos^{2} (x) \sin (y)$ .

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

Step 4.3.3

Factor $\sin (y)$ out of $\sin (y) \sin^{2} (x) + \sin (y) \cos^{2} (x)$ .

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

Step 4.4

Apply pythagorean identity.

$\sin (y) \cdot 1$

Step 4.5

Multiply $\sin (y)$ by $1$ .

$\sin (y)$

Step 5

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

$\sin (x) \cos (x - y) - \cos (x) \sin (x - y) = \sin (y)$ is an identity