Trigonometry Examples

$\tan (x + y) - \tan (x)$

Step 1

Apply the sum of angles identity.

$\frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)} - \tan (x)$

Step 2

To write $- \tan (x)$ as a fraction with a common denominator, multiply by $\frac{1 - \tan (x) \tan (y)}{1 - \tan (x) \tan (y)}$ .

$\frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)} - \tan (x) \cdot \frac{1 - \tan (x) \tan (y)}{1 - \tan (x) \tan (y)}$

Step 3

Simplify terms.

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Step 3.1

Combine $- \tan (x)$ and $\frac{1 - \tan (x) \tan (y)}{1 - \tan (x) \tan (y)}$ .

$\frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)} + \frac{- \tan (x) (1 - \tan (x) \tan (y))}{1 - \tan (x) \tan (y)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\tan (x) + \tan (y) - \tan (x) (1 - \tan (x) \tan (y))}{1 - \tan (x) \tan (y)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{\tan (x) + \tan (y) - \tan (x) \cdot 1 - \tan (x) (- \tan (x) \tan (y))}{1 - \tan (x) \tan (y)}$

Step 4.2

Multiply $- 1$ by $1$ .

$\frac{\tan (x) + \tan (y) - \tan (x) - \tan (x) (- \tan (x) \tan (y))}{1 - \tan (x) \tan (y)}$

Step 4.3

Multiply $- \tan (x) (- \tan (x) \tan (y))$ .

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\tan (x) + \tan (y) - \tan (x) + 1 \tan (x) (\tan (x) \tan (y))}{1 - \tan (x) \tan (y)}$

Step 4.3.2

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

$\frac{\tan (x) + \tan (y) - \tan (x) + \tan (x) (\tan (x) \tan (y))}{1 - \tan (x) \tan (y)}$

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Add $1$ and $1$ .

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

Step 4.4

Subtract $\tan (x)$ from $\tan (x)$ .

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

Step 4.5

Add $0$ and $\tan (y)$ .

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

Step 4.6

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

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Step 4.6.1

Multiply by $1$ .

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

Step 4.6.2

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

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

Step 4.6.3

Factor $\tan (y)$ out of $\tan (y) \cdot 1 + \tan (y) \tan^{2} (x)$ .

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

Step 4.7

Rearrange terms.

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

Step 4.8

Apply pythagorean identity.

$\frac{\tan (y) \sec^{2} (x)}{1 - \tan (x) \tan (y)}$