Trigonometry Examples

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

Step 1

Start on the left side.

$\cos (x - \frac{π}{4})$

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

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

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

Step 5

Now consider the right side of the equation.

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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