Precalculus Examples

$\tan (2 π + θ) = \tan (θ)$

Step 1

Start on the left side.

$\tan (2 π + θ)$

Step 2

Apply the sum of angles identity.

$\frac{\tan (2 π) + \tan (θ)}{1 - \tan (2 π) \tan (θ)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\tan (0) + \tan (θ)}{1 - \tan (2 π) \tan (θ)}$

Step 3.1.2

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

$\frac{0 + \tan (θ)}{1 - \tan (2 π) \tan (θ)}$

Step 3.1.3

Add $0$ and $\tan (θ)$ .

$\frac{\tan (θ)}{1 - \tan (2 π) \tan (θ)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 3.2.2

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

$\frac{\tan (θ)}{1 - 0 \tan (θ)}$

Step 3.2.3

Multiply $- 1$ by $0$ .

$\frac{\tan (θ)}{1 + 0 \tan (θ)}$

Step 3.2.4

Multiply $0$ by $\tan (θ)$ .

$\frac{\tan (θ)}{1 + 0}$

Step 3.2.5

Add $1$ and $0$ .

$\frac{\tan (θ)}{1}$

Step 3.3

Divide $\tan (θ)$ by $1$ .

$\tan (θ)$

Step 4

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

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