Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Apply the difference of angles identity.

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

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.

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Add $1$ and $1$ .

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

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

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Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.4.2

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

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

Step 4.1.4.3

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

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

Step 4.1.4.4

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

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

Step 4.1.4.5

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

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

Step 4.1.4.6

Add $1$ and $1$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.4

Rearrange terms.

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

Step 4.5

Apply pythagorean identity.

$\cos (x) \cdot 1$

Step 4.6

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

$\cos (x)$

Step 5

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

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