Precalculus Examples

$\cos (\frac{π}{3} + x) + \cos (\frac{π}{3} - x) = \cos (x)$

Step 1

Start on the left side.

$\cos (\frac{π}{3} + x) + \cos (\frac{π}{3} - x)$

Step 2

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

$\cos (\frac{π}{3}) \cos (x) - \sin (\frac{π}{3}) \sin (x) + \cos (\frac{π}{3} - x)$

Step 3

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

$\cos (\frac{π}{3}) \cos (x) - \sin (\frac{π}{3}) \sin (x) + \cos (\frac{π}{3}) \cos (- x) - \sin (\frac{π}{3}) \sin (- x)$

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{1}{2} \cos (x) - \sin (\frac{π}{3}) \sin (x) + \cos (\frac{π}{3}) \cos (- x) - \sin (\frac{π}{3}) \sin (- x)$

Step 4.1.2

Combine $\frac{1}{2}$ and $\cos (x)$ .

$\frac{\cos (x)}{2} - \sin (\frac{π}{3}) \sin (x) + \cos (\frac{π}{3}) \cos (- x) - \sin (\frac{π}{3}) \sin (- x)$

Step 4.1.3

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\cos (x)}{2} - \frac{\sqrt{3}}{2} \sin (x) + \cos (\frac{π}{3}) \cos (- x) - \sin (\frac{π}{3}) \sin (- x)$

Step 4.1.4

Combine $\sin (x)$ and $\frac{\sqrt{3}}{2}$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \cos (\frac{π}{3}) \cos (- x) - \sin (\frac{π}{3}) \sin (- x)$

Step 4.1.5

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{1}{2} \cos (- x) - \sin (\frac{π}{3}) \sin (- x)$

Step 4.1.6

Since $\cos (- x)$ is an even function, rewrite $\cos (- x)$ as $\cos (x)$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{1}{2} \cos (x) - \sin (\frac{π}{3}) \sin (- x)$

Step 4.1.7

Combine $\frac{1}{2}$ and $\cos (x)$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} - \sin (\frac{π}{3}) \sin (- x)$

Step 4.1.8

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} - \frac{\sqrt{3}}{2} \sin (- x)$

Step 4.1.9

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} - \frac{\sqrt{3}}{2} (- \sin (x))$

Step 4.1.10

Multiply $- \frac{\sqrt{3}}{2} (- \sin (x))$ .

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Step 4.1.10.1

Multiply $- 1$ by $- 1$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} + 1 \frac{\sqrt{3}}{2} \sin (x)$

Step 4.1.10.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} + \frac{\sqrt{3}}{2} \sin (x)$

Step 4.1.10.3

Combine $\frac{\sqrt{3}}{2}$ and $\sin (x)$ .

$\frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} + \frac{\sqrt{3} \sin (x)}{2}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{\cos (x) - \sin (x) \sqrt{3} + \cos (x) + \sqrt{3} \sin (x)}{2}$

Step 4.3

Combine the opposite terms in $\cos (x) - \sin (x) \sqrt{3} + \cos (x) + \sqrt{3} \sin (x)$ .

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Step 4.3.1

Reorder the factors in the terms $- \sin (x) \sqrt{3}$ and $\sqrt{3} \sin (x)$ .

$\frac{\cos (x) - \sqrt{3} \sin (x) + \cos (x) + \sqrt{3} \sin (x)}{2}$

Step 4.3.2

Add $- \sqrt{3} \sin (x)$ and $\sqrt{3} \sin (x)$ .

$\frac{\cos (x) + 0 + \cos (x)}{2}$

Step 4.3.3

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

$\frac{\cos (x) + \cos (x)}{2}$

Step 4.4

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

$\frac{2 \cos (x)}{2}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

$\frac{2 \cos (x)}{2}$

Step 4.5.2

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

$\cos (x)$

Step 5

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

$\cos (\frac{π}{3} + x) + \cos (\frac{π}{3} - x) = \cos (x)$ is an identity