Trigonometry Examples

$\cos (π + x) = - \cos (x)$

Step 1

Start on the left side.

$\cos (π + x)$

Step 2

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

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

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.

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

Step 3.1.2

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

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

Step 3.1.3

Multiply $- 1$ by $1$ .

$- 1 \cos (x) - \sin (π) \sin (x)$

Step 3.1.4

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

$- \cos (x) - \sin (π) \sin (x)$

Step 3.1.5

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

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

Step 3.1.6

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

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

Step 3.1.7

Multiply $- 1$ by $0$ .

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

Step 3.1.8

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.

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