Trigonometry Examples

$\cos (x - π) = - \cos (x)$

Step 1

Start on the left side.

$\cos (x - π)$

Step 2

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify each term.

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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 cosine is negative in the second quadrant.

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.4

Move $- 1$ to the left of $\cos (x)$ .

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

Step 3.1.5

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$- \cos (x) + \sin (x) \sin (π)$

Step 3.1.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 3.1.7

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

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

Step 3.1.8

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

$- \cos (x) + 0$

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.

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