Trigonometry Examples

$\tan (x + π) = \tan (x)$

Step 1

Start on the left side.

$\tan (x + π)$

Step 2

Apply the sum of angles identity.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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.

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $- 1$ by $0$ .

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

Step 3.1.4

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

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

Step 3.2

Simplify the denominator.

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Step 3.2.1

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.

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $- 1$ by $0$ .

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

Step 3.2.4

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

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Step 3.2.4.1

Multiply $0$ by $- 1$ .

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

Step 3.2.4.2

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

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

Step 3.2.5

Add $1$ and $0$ .

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

Step 3.3

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

$\tan (x)$

Step 4

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

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