Trigonometry Examples

$\tan (x + π) - \tan (π - x) = 2 \tan (x)$

Step 1

Start on the left side.

$\tan (x + π) - \tan (π - x)$

Step 2

Apply the sum of angles identity.

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

Step 3

Apply the sum of angles identity.

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

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

Simplify the numerator.

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Step 4.1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{\tan (x) - \tan (0)}{1 - \tan (x) \tan (π)} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.1.2

The exact value of $\tan (0)$ is $0$ .

$\frac{\tan (x) - 0}{1 - \tan (x) \tan (π)} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.1.3

Multiply $- 1$ by $0$ .

$\frac{\tan (x) + 0}{1 - \tan (x) \tan (π)} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.1.4

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

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

Step 4.1.2

Simplify the denominator.

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Step 4.1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{\tan (x)}{1 - \tan (x) (- \tan (0))} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.2.2

The exact value of $\tan (0)$ is $0$ .

$\frac{\tan (x)}{1 - \tan (x) (- 0)} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.2.3

Multiply $- 1$ by $0$ .

$\frac{\tan (x)}{1 - \tan (x) \cdot 0} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.2.4

Multiply $- \tan (x) \cdot 0$ .

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Step 4.1.2.4.1

Multiply $0$ by $- 1$ .

$\frac{\tan (x)}{1 + 0 \tan (x)} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.2.4.2

Multiply $0$ by $\tan (x)$ .

$\frac{\tan (x)}{1 + 0} - \frac{\tan (π) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.2.5

Add $1$ and $0$ .

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

Step 4.1.3

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

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

Step 4.1.4

Simplify the numerator.

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Step 4.1.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\tan (x) - \frac{- \tan (0) + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.4.2

The exact value of $\tan (0)$ is $0$ .

$\tan (x) - \frac{- 0 + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.4.3

Multiply $- 1$ by $0$ .

$\tan (x) - \frac{0 + \tan (- x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.4.4

Since $\tan (- x)$ is an odd function, rewrite $\tan (- x)$ as $- \tan (x)$ .

$\tan (x) - \frac{0 - \tan (x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.4.5

Subtract $\tan (x)$ from $0$ .

$\tan (x) - \frac{- \tan (x)}{1 - \tan (π) \tan (- x)}$

Step 4.1.5

Simplify the denominator.

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Step 4.1.5.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\tan (x) - \frac{- \tan (x)}{1 - - \tan (0) \tan (- x)}$

Step 4.1.5.2

The exact value of $\tan (0)$ is $0$ .

$\tan (x) - \frac{- \tan (x)}{1 - - 0 \tan (- x)}$

Step 4.1.5.3

Multiply $- - 0$ .

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Step 4.1.5.3.1

Multiply $- 1$ by $0$ .

$\tan (x) - \frac{- \tan (x)}{1 - 0 \tan (- x)}$

Step 4.1.5.3.2

Multiply $- 1$ by $0$ .

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

Step 4.1.5.4

Since $\tan (- x)$ is an odd function, rewrite $\tan (- x)$ as $- \tan (x)$ .

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

Step 4.1.5.5

Multiply $0 (- \tan (x))$ .

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Step 4.1.5.5.1

Multiply $- 1$ by $0$ .

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

Step 4.1.5.5.2

Multiply $0$ by $\tan (x)$ .

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

Step 4.1.5.6

Add $1$ and $0$ .

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

Step 4.1.6

Divide $- \tan (x)$ by $1$ .

$\tan (x) - - \tan (x)$

Step 4.1.7

Multiply $- - \tan (x)$ .

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Step 4.1.7.1

Multiply $- 1$ by $- 1$ .

$\tan (x) + 1 \tan (x)$

Step 4.1.7.2

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

$\tan (x) + \tan (x)$

Step 4.2

Add $\tan (x)$ and $\tan (x)$ .

$2 \tan (x)$

Step 5

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

$\tan (x + π) - \tan (π - x) = 2 \tan (x)$ is an identity