Algebra Examples

$\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)}$

Step 1

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

$\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)} \cdot \frac{\cos (A) - \cos (B)}{\cos (A) - \cos (B)} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)}$

Step 2

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

$\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)} \cdot \frac{\cos (A) - \cos (B)}{\cos (A) - \cos (B)} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)} \cdot \frac{\sin (A) - \sin (B)}{\sin (A) - \sin (B)}$

Step 3

Write each expression with a common denominator of $(\sin (A) - \sin (B)) (\cos (A) - \cos (B))$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{\cos (A) + \cos (B)}{\sin (A) - \sin (B)}$ by $\frac{\cos (A) - \cos (B)}{\cos (A) - \cos (B)}$ .

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B))}{(\sin (A) - \sin (B)) (\cos (A) - \cos (B))} + \frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)} \cdot \frac{\sin (A) - \sin (B)}{\sin (A) - \sin (B)}$

Step 3.2

Multiply $\frac{\sin (A) + \sin (B)}{\cos (A) - \cos (B)}$ by $\frac{\sin (A) - \sin (B)}{\sin (A) - \sin (B)}$ .

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B))}{(\sin (A) - \sin (B)) (\cos (A) - \cos (B))} + \frac{(\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 3.3

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

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))} + \frac{(\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 4

Combine the numerators over the common denominator.

$\frac{(\cos (A) + \cos (B)) (\cos (A) - \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(\cos (A) + \cos (B)) (\cos (A) - \cos (B))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{\cos (A) (\cos (A) - \cos (B)) + \cos (B) (\cos (A) - \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.1.2

Apply the distributive property.

$\frac{\cos (A) \cos (A) + \cos (A) (- \cos (B)) + \cos (B) (\cos (A) - \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.1.3

Apply the distributive property.

$\frac{\cos (A) \cos (A) + \cos (A) (- \cos (B)) + \cos (B) \cos (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.2

Combine the opposite terms in $\cos (A) \cos (A) + \cos (A) (- \cos (B)) + \cos (B) \cos (A) + \cos (B) (- \cos (B))$ .

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Step 5.2.1

Reorder the factors in the terms $\cos (A) (- \cos (B))$ and $\cos (B) \cos (A)$ .

$\frac{\cos (A) \cos (A) - \cos (A) \cos (B) + \cos (A) \cos (B) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.2.2

Add $- \cos (A) \cos (B)$ and $\cos (A) \cos (B)$ .

$\frac{\cos (A) \cos (A) + 0 + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.2.3

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

$\frac{\cos (A) \cos (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $\cos (A) \cos (A)$ .

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Step 5.3.1.1

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

$\frac{\cos^{1} (A) \cos (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.1.2

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

$\frac{\cos^{1} (A) \cos^{1} (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.1.3

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

$\frac{{\cos (A)}^{1 + 1} + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.1.4

Add $1$ and $1$ .

$\frac{\cos^{2} (A) + \cos (B) (- \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.2

Rewrite using the commutative property of multiplication.

$\frac{\cos^{2} (A) - \cos (B) \cos (B) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.3

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

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Step 5.3.3.1

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

$\frac{\cos^{2} (A) - (\cos^{1} (B) \cos (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.3.2

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

$\frac{\cos^{2} (A) - (\cos^{1} (B) \cos^{1} (B)) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.3.3

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

$\frac{\cos^{2} (A) - {\cos (B)}^{1 + 1} + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.3.3.4

Add $1$ and $1$ .

$\frac{\cos^{2} (A) - \cos^{2} (B) + (\sin (A) + \sin (B)) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.4

Expand $(\sin (A) + \sin (B)) (\sin (A) - \sin (B))$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin (A) (\sin (A) - \sin (B)) + \sin (B) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.4.2

Apply the distributive property.

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin (A) \sin (A) + \sin (A) (- \sin (B)) + \sin (B) (\sin (A) - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.4.3

Apply the distributive property.

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin (A) \sin (A) + \sin (A) (- \sin (B)) + \sin (B) \sin (A) + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.5

Combine the opposite terms in $\sin (A) \sin (A) + \sin (A) (- \sin (B)) + \sin (B) \sin (A) + \sin (B) (- \sin (B))$ .

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Step 5.5.1

Reorder the factors in the terms $\sin (A) (- \sin (B))$ and $\sin (B) \sin (A)$ .

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin (A) \sin (A) - \sin (A) \sin (B) + \sin (A) \sin (B) + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.5.2

Add $- \sin (A) \sin (B)$ and $\sin (A) \sin (B)$ .

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin (A) \sin (A) + 0 + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.5.3

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

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin (A) \sin (A) + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6

Simplify each term.

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Step 5.6.1

Multiply $\sin (A) \sin (A)$ .

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Step 5.6.1.1

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

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{1} (A) \sin (A) + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.1.2

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

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{1} (A) \sin^{1} (A) + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.1.3

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

$\frac{\cos^{2} (A) - \cos^{2} (B) + {\sin (A)}^{1 + 1} + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.1.4

Add $1$ and $1$ .

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{2} (A) + \sin (B) (- \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.2

Rewrite using the commutative property of multiplication.

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{2} (A) - \sin (B) \sin (B)}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.3

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

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Step 5.6.3.1

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

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{2} (A) - (\sin^{1} (B) \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.3.2

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

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{2} (A) - (\sin^{1} (B) \sin^{1} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.3.3

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

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{2} (A) - {\sin (B)}^{1 + 1}}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.6.3.4

Add $1$ and $1$ .

$\frac{\cos^{2} (A) - \cos^{2} (B) + \sin^{2} (A) - \sin^{2} (B)}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7

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

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Step 5.7.1

Regroup terms.

$\frac{- \cos^{2} (B) + \cos^{2} (A) + \sin^{2} (A) - \sin^{2} (B)}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.2

Rearrange terms.

$\frac{- \cos^{2} (B) + \sin^{2} (A) + \cos^{2} (A) - \sin^{2} (B)}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.3

Apply pythagorean identity.

$\frac{- \cos^{2} (B) + 1 - \sin^{2} (B)}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.4

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

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Step 5.7.4.1

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

$\frac{- \cos^{2} (B) + 1^{2} - \sin^{2} (B)}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 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 $b = \sin (B)$ .

$\frac{- \cos^{2} (B) + (1 + \sin (B)) (1 - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.5

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

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Step 5.7.5.1

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

$\frac{- (\cos^{2} (B)) + (1 + \sin (B)) (1 - \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.5.2

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

$\frac{- (\cos^{2} (B)) - ((- 1 - \sin (B)) (1 - \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.5.3

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

$\frac{- (\cos^{2} (B) + (- 1 - \sin (B)) (1 - \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.6

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

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Step 5.7.6.1

Apply the distributive property.

$\frac{- (\cos^{2} (B) - 1 (1 - \sin (B)) - \sin (B) (1 - \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.6.2

Apply the distributive property.

$\frac{- (\cos^{2} (B) - 1 \cdot 1 - 1 (- \sin (B)) - \sin (B) (1 - \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.6.3

Apply the distributive property.

$\frac{- (\cos^{2} (B) - 1 \cdot 1 - 1 (- \sin (B)) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7

Simplify and combine like terms.

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Step 5.7.7.1

Simplify each term.

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Step 5.7.7.1.1

Multiply $- 1$ by $1$ .

$\frac{- (\cos^{2} (B) - 1 - 1 (- \sin (B)) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.2

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

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Step 5.7.7.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- (\cos^{2} (B) - 1 + 1 \sin (B) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.2.2

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

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) \cdot 1 - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.3

Multiply $- 1$ by $1$ .

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) - \sin (B) (- \sin (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.4

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

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Step 5.7.7.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) + 1 \sin (B) \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.4.2

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

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) + \sin (B) \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.4.3

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

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) + \sin^{1} (B) \sin (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.4.4

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

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) + \sin^{1} (B) \sin^{1} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.4.5

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

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) + {\sin (B)}^{1 + 1})}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.1.4.6

Add $1$ and $1$ .

$\frac{- (\cos^{2} (B) - 1 + \sin (B) - \sin (B) + \sin^{2} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.2

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

$\frac{- (\cos^{2} (B) - 1 + 0 + \sin^{2} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.7.3

Add $- 1$ and $0$ .

$\frac{- (\cos^{2} (B) - 1 + \sin^{2} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.8

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

$\frac{- (\cos^{2} (B) - 1 (1) + \sin^{2} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.9

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

$\frac{- (\cos^{2} (B) - 1 (1) - 1 (- \sin^{2} (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.10

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

$\frac{- (\cos^{2} (B) - 1 (1 - \sin^{2} (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.11

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

$\frac{- (\cos^{2} (B) - (1 - \sin^{2} (B)))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.12

Apply pythagorean identity.

$\frac{- (\cos^{2} (B) - \cos^{2} (B))}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 5.7.13

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

$\frac{- 0}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 6

Multiply by zero.

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Step 6.1

Multiply $- 1$ by $0$ .

$\frac{0}{(\cos (A) - \cos (B)) (\sin (A) - \sin (B))}$

Step 6.2

Divide $0$ by $(\cos (A) - \cos (B)) (\sin (A) - \sin (B))$ .

$0$