Trigonometry Examples

$\cos (x - 2 π) = \cos (x)$

Step 1

Start on the left side.

$\cos (x - 2 π)$

Step 2

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$\cos (x) \cos (- 2 π) - \sin (x) \sin (- 2 π)$

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$\cos (x) \cos (0) - \sin (x) \sin (- 2 π)$

Step 3.1.2

The exact value of $\cos (0)$ is $1$ .

$\cos (x) \cdot 1 - \sin (x) \sin (- 2 π)$

Step 3.1.3

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

$\cos (x) - \sin (x) \sin (- 2 π)$

Step 3.1.4

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

$\cos (x) - \sin (x) \sin (0)$

Step 3.1.5

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

$\cos (x) - \sin (x) \cdot 0$

Step 3.1.6

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

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Step 3.1.6.1

Multiply $0$ by $- 1$ .

$\cos (x) + 0 \sin (x)$

Step 3.1.6.2

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

$\cos (x) + 0$

Step 3.2

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

$\cos (x)$

Step 4

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

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