Trigonometry Examples

$\cos (x + \frac{π}{4}) + \cos (x - \frac{π}{4}) = \sqrt{2} \cos (x)$

Step 1

Start on the left side.

$\cos (x + \frac{π}{4}) + \cos (x - \frac{π}{4})$

Step 2

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

$\cos (x) \cos (\frac{π}{4}) - \sin (x) \sin (\frac{π}{4}) + \cos (x - \frac{π}{4})$

Step 3

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

$\cos (x) \cos (\frac{π}{4}) - \sin (x) \sin (\frac{π}{4}) + \cos (x) \cos (- \frac{π}{4}) - \sin (x) \sin (- \frac{π}{4})$

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 $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\cos (x) \frac{\sqrt{2}}{2} - \sin (x) \sin (\frac{π}{4}) + \cos (x) \cos (- \frac{π}{4}) - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.2

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

$\frac{\cos (x) \sqrt{2}}{2} - \sin (x) \sin (\frac{π}{4}) + \cos (x) \cos (- \frac{π}{4}) - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\cos (x) \sqrt{2}}{2} - \sin (x) \frac{\sqrt{2}}{2} + \cos (x) \cos (- \frac{π}{4}) - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.4

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

$\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \cos (x) \cos (- \frac{π}{4}) - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.5

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

$\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \cos (x) \cos (\frac{7 π}{4}) - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.6

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

$\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \cos (x) \cos (\frac{π}{4}) - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.7

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \cos (x) \frac{\sqrt{2}}{2} - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.8

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

$\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \frac{\cos (x) \sqrt{2}}{2} - \sin (x) \sin (- \frac{π}{4})$

Step 4.1.9

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

$\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \frac{\cos (x) \sqrt{2}}{2} - \sin (x) \sin (\frac{7 π}{4})$

Step 4.1.10

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \frac{\cos (x) \sqrt{2}}{2} - \sin (x) (- \sin (\frac{π}{4}))$

Step 4.1.11

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.1.12

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

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Step 4.1.12.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.12.2

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

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

Step 4.1.12.3

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

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

Step 4.2

Combine the opposite terms in $\frac{\cos (x) \sqrt{2}}{2} - \frac{\sqrt{2} \sin (x)}{2} + \frac{\cos (x) \sqrt{2}}{2} + \frac{\sin (x) \sqrt{2}}{2}$ .

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Step 4.2.1

Reorder the factors in the terms $- \frac{\sqrt{2} \sin (x)}{2}$ and $\frac{\sin (x) \sqrt{2}}{2}$ .

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

Step 4.2.2

Add $- \frac{\sqrt{2} \sin (x)}{2}$ and $\frac{\sqrt{2} \sin (x)}{2}$ .

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

Step 4.2.3

Add $\frac{\cos (x) \sqrt{2}}{2}$ and $0$ .

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Add $\cos (x) \sqrt{2}$ and $\cos (x) \sqrt{2}$ .

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

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Divide $\cos (x) \sqrt{2}$ by $1$ .

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

Step 5

Reorder factors in $\cos (x) \sqrt{2}$ .

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

Step 6

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

$\cos (x + \frac{π}{4}) + \cos (x - \frac{π}{4}) = \sqrt{2} \cos (x)$ is an identity