Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the sum of angles identity.

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

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.3

Cancel the common factor of $\tan (A) + \tan (B)$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 1.4

Apply the difference of angles identity.

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

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.6

Cancel the common factor of $\tan (A) - \tan (B)$ .

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Step 1.6.1

Cancel the common factor.

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

Step 1.6.2

Rewrite the expression.

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $1 - \tan (A) \tan (B) + 1 + \tan (A) \tan (B)$ .

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Step 2.1.1

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

$1 + 0 + 1$

Step 2.1.2

Add $1$ and $0$ .

$1 + 1$

Step 2.2

Add $1$ and $1$ .

$2$