Expand Using Sum/Difference Formulas cos(45)

$\cos (45)$

The angle $45$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\cos (45 + 0)$

Use the sum formula for cosine to simplify the expression. The formula states that $\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))$ .

$\cos (0) \cdot \cos (45) - \sin (0) \cdot \sin (45)$

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

$(1) \cdot \cos (45) - \sin (0) \cdot \sin (45)$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$(1) \cdot (\frac{\sqrt{2}}{2}) - \sin (0) \cdot \sin (45)$

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

$(1) \cdot (\frac{\sqrt{2}}{2}) + 0 \cdot \sin (45)$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$(1) \cdot (\frac{\sqrt{2}}{2}) + 0 \cdot \frac{\sqrt{2}}{2}$

Simplify each term.

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

$\frac{\sqrt{2}}{2} + 0 \cdot \frac{\sqrt{2}}{2}$

Multiply $0$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} + 0$

Add $\frac{\sqrt{2}}{2}$ and $0$ .

$\frac{\sqrt{2}}{2}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$

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