Trigonometry Examples

\frac{tan (A) + tan (B)}{tan (A + B)} + \frac{tan (A) - tan (B)}{tan (A - B)}

$\frac{\tan (A) + \tan (B)}{\tan (A + B)} + \frac{\tan (A) - \tan (B)}{\tan (A - B)}$

Step 1

Simplify each term.

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Step 1.1

Apply the sum of angles identity.

\frac{tan (A) + tan (B)}{\frac{tan (A) + tan (B)}{1 - tan (A) tan (B)}} + \frac{tan (A) - tan (B)}{tan (A - B)}

$\frac{\tan (A) + \tan (B)}{\frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}} + \frac{\tan (A) - \tan (B)}{\tan (A - B)}$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

(tan (A) + tan (B)) \frac{1 - tan (A) tan (B)}{tan (A) + tan (B)} + \frac{tan (A) - tan (B)}{tan (A - B)}

$(\tan (A) + \tan (B)) \frac{1 - \tan (A) \tan (B)}{\tan (A) + \tan (B)} + \frac{\tan (A) - \tan (B)}{\tan (A - B)}$

Step 1.3

Cancel the common factor of

tan (A) + tan (B)

$\tan (A) + \tan (B)$ .

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Step 1.3.1

Cancel the common factor.

(tan (A) + tan (B)) \frac{1 - tan (A) tan (B)}{tan (A) + tan (B)} + \frac{tan (A) - tan (B)}{tan (A - B)}

$(\tan (A) + \tan (B)) \frac{1 - \tan (A) \tan (B)}{\tan (A) + \tan (B)} + \frac{\tan (A) - \tan (B)}{\tan (A - B)}$

Step 1.3.2

Rewrite the expression.

1 - tan (A) tan (B) + \frac{tan (A) - tan (B)}{tan (A - B)}

$1 - \tan (A) \tan (B) + \frac{\tan (A) - \tan (B)}{\tan (A - B)}$

1 - tan (A) tan (B) + \frac{tan (A) - tan (B)}{tan (A - B)}

$1 - \tan (A) \tan (B) + \frac{\tan (A) - \tan (B)}{\tan (A - B)}$

Step 1.4

Apply the difference of angles identity.

1 - tan (A) tan (B) + \frac{tan (A) - tan (B)}{\frac{tan (A) - tan (B)}{1 + tan (A) tan (B)}}

$1 - \tan (A) \tan (B) + \frac{\tan (A) - \tan (B)}{\frac{\tan (A) - \tan (B)}{1 + \tan (A) \tan (B)}}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

1 - tan (A) tan (B) + (tan (A) - tan (B)) \frac{1 + tan (A) tan (B)}{tan (A) - tan (B)}

$1 - \tan (A) \tan (B) + (\tan (A) - \tan (B)) \frac{1 + \tan (A) \tan (B)}{\tan (A) - \tan (B)}$

Step 1.6

Cancel the common factor of

tan (A) - tan (B)

$\tan (A) - \tan (B)$ .

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Step 1.6.1

Cancel the common factor.

1 - tan (A) tan (B) + (tan (A) - tan (B)) \frac{1 + tan (A) tan (B)}{tan (A) - tan (B)}

$1 - \tan (A) \tan (B) + (\tan (A) - \tan (B)) \frac{1 + \tan (A) \tan (B)}{\tan (A) - \tan (B)}$

Step 1.6.2

Rewrite the expression.

1 - tan (A) tan (B) + 1 + tan (A) tan (B)

$1 - \tan (A) \tan (B) + 1 + \tan (A) \tan (B)$

1 - tan (A) tan (B) + 1 + tan (A) tan (B)

$1 - \tan (A) \tan (B) + 1 + \tan (A) \tan (B)$

1 - tan (A) tan (B) + 1 + tan (A) tan (B)

$1 - \tan (A) \tan (B) + 1 + \tan (A) \tan (B)$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in

1 - tan (A) tan (B) + 1 + tan (A) tan (B)

$1 - \tan (A) \tan (B) + 1 + \tan (A) \tan (B)$ .

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Step 2.1.1

Add

- tan (A) tan (B)

$- \tan (A) \tan (B)$ and

tan (A) tan (B)

$\tan (A) \tan (B)$ .

1 + 0 + 1

$1 + 0 + 1$

Step 2.1.2

Add

1

$1$ and

0

$0$ .

1 + 1

$1 + 1$

1 + 1

$1 + 1$

Step 2.2

Add

1

$1$ and

1

$1$ .

2

$2$

2

$2$