Trigonometry Examples

$\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)} + \frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)} = 0$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

To write $\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x) + \cos (y)}{\cos (x) + \cos (y)}$ .

$\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)} \cdot \frac{\cos (x) + \cos (y)}{\cos (x) + \cos (y)} + \frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)}$

Step 2.2

To write $\frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x) + \sin (y)}{\sin (x) + \sin (y)}$ .

$\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)} \cdot \frac{\cos (x) + \cos (y)}{\cos (x) + \cos (y)} + \frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)} \cdot \frac{\sin (x) + \sin (y)}{\sin (x) + \sin (y)}$

Step 2.3

Write each expression with a common denominator of $(\sin (x) + \sin (y)) (\cos (x) + \cos (y))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)}$ by $\frac{\cos (x) + \cos (y)}{\cos (x) + \cos (y)}$ .

$\frac{(\cos (x) - \cos (y)) (\cos (x) + \cos (y))}{(\sin (x) + \sin (y)) (\cos (x) + \cos (y))} + \frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)} \cdot \frac{\sin (x) + \sin (y)}{\sin (x) + \sin (y)}$

Step 2.3.2

Multiply $\frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)}$ by $\frac{\sin (x) + \sin (y)}{\sin (x) + \sin (y)}$ .

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

Apply the distributive property.

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

Step 2.5.1.2

Apply the distributive property.

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

Step 2.5.1.3

Apply the distributive property.

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

Step 2.5.2

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

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Step 2.5.2.1

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

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

Step 2.5.2.2

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

$\frac{\cos (x) \cos (x) + 0 - \cos (y) \cos (y) + (\sin (x) - \sin (y)) (\sin (x) + \sin (y))}{(\cos (x) + \cos (y)) (\sin (x) + \sin (y))}$

Step 2.5.2.3

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

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

Step 2.5.3

Simplify each term.

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Step 2.5.3.1

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

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Step 2.5.3.1.1

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

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

Step 2.5.3.1.2

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

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

Step 2.5.3.1.3

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

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

Step 2.5.3.1.4

Add $1$ and $1$ .

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

Step 2.5.3.2

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

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Step 2.5.3.2.1

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

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

Step 2.5.3.2.2

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

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

Step 2.5.3.2.3

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

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

Step 2.5.3.2.4

Add $1$ and $1$ .

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

Step 2.5.4

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

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Step 2.5.4.1

Apply the distributive property.

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

Step 2.5.4.2

Apply the distributive property.

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

Step 2.5.4.3

Apply the distributive property.

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

Step 2.5.5

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

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Step 2.5.5.1

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

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

Step 2.5.5.2

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

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

Step 2.5.5.3

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

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

Step 2.5.6

Simplify each term.

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Step 2.5.6.1

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

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Step 2.5.6.1.1

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

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

Step 2.5.6.1.2

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

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

Step 2.5.6.1.3

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

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

Step 2.5.6.1.4

Add $1$ and $1$ .

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

Step 2.5.6.2

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

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Step 2.5.6.2.1

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

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

Step 2.5.6.2.2

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

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

Step 2.5.6.2.3

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

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

Step 2.5.6.2.4

Add $1$ and $1$ .

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

Step 2.5.7

Rewrite $\cos^{2} (x) - \cos^{2} (y) + \sin^{2} (x) - \sin^{2} (y)$ in a factored form.

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Step 2.5.7.1

Regroup terms.

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

Step 2.5.7.2

Rearrange terms.

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

Step 2.5.7.3

Apply pythagorean identity.

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

Step 2.5.7.4

Rewrite $1 - \sin^{2} (y)$ in a factored form.

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Step 2.5.7.4.1

Rewrite $1$ as $1^{2}$ .

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

Step 2.5.7.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (y)$ .

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

Step 2.5.7.5

Factor $- 1$ out of $- \cos^{2} (y) + (1 + \sin (y)) (1 - \sin (y))$ .

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Step 2.5.7.5.1

Factor $- 1$ out of $- \cos^{2} (y)$ .

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

Step 2.5.7.5.2

Factor $- 1$ out of $(1 + \sin (y)) (1 - \sin (y))$ .

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

Step 2.5.7.5.3

Factor $- 1$ out of $- (\cos^{2} (y)) - ((- 1 - \sin (y)) (1 - \sin (y)))$ .

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

Step 2.5.7.6

Expand $(- 1 - \sin (y)) (1 - \sin (y))$ using the FOIL Method.

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Step 2.5.7.6.1

Apply the distributive property.

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

Step 2.5.7.6.2

Apply the distributive property.

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

Step 2.5.7.6.3

Apply the distributive property.

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

Step 2.5.7.7

Simplify and combine like terms.

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Step 2.5.7.7.1

Simplify each term.

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Step 2.5.7.7.1.1

Multiply $- 1$ by $1$ .

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

Step 2.5.7.7.1.2

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

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Step 2.5.7.7.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.7.7.1.2.2

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

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

Step 2.5.7.7.1.3

Multiply $- 1$ by $1$ .

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

Step 2.5.7.7.1.4

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

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Step 2.5.7.7.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.5.7.7.1.4.2

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

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

Step 2.5.7.7.1.4.3

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

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

Step 2.5.7.7.1.4.4

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

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

Step 2.5.7.7.1.4.5

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

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

Step 2.5.7.7.1.4.6

Add $1$ and $1$ .

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

Step 2.5.7.7.2

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

$\frac{- (\cos^{2} (y) - 1 + 0 + \sin^{2} (y))}{(\cos (x) + \cos (y)) (\sin (x) + \sin (y))}$

Step 2.5.7.7.3

Add $- 1$ and $0$ .

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

Step 2.5.7.8

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.5.7.9

Factor $- 1$ out of $\sin^{2} (y)$ .

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

Step 2.5.7.10

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (y))$ .

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

Step 2.5.7.11

Rewrite $- 1 (1 - \sin^{2} (y))$ as $- (1 - \sin^{2} (y))$ .

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

Step 2.5.7.12

Apply pythagorean identity.

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

Step 2.5.7.13

Subtract $\cos^{2} (y)$ from $\cos^{2} (y)$ .

$\frac{- 0}{(\cos (x) + \cos (y)) (\sin (x) + \sin (y))}$

Step 2.6

Multiply $- 1$ by $0$ .

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

Step 2.7

Divide $0$ by $(\cos (x) + \cos (y)) (\sin (x) + \sin (y))$ .

$0$

Step 3

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

$\frac{\cos (x) - \cos (y)}{\sin (x) + \sin (y)} + \frac{\sin (x) - \sin (y)}{\cos (x) + \cos (y)} = 0$ is an identity