Find the Exact Value cos(60 degrees )cos(45 degrees )-sin(60 degrees )sin(45 degrees )

$\cos (60 °) \cos (45 °) - \sin (60 °) \sin (45 °)$

Step 1

The exact value of $\cos (60 °)$ is $\frac{1}{2}$ .

$\frac{1}{2} \cos (45 °) - \sin (60 °) \sin (45 °)$

Step 2

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

$\frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \sin (60 °) \sin (45 °)$

Step 3

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

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

$\frac{\sqrt{2}}{2 \cdot 2} - \sin (60 °) \sin (45 °)$

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{4} - \sin (60 °) \sin (45 °)$

Step 4

The exact value of $\sin (60 °)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{2}}{4} - \frac{\sqrt{3}}{2} \sin (45 °)$

Step 5

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

$\frac{\sqrt{2}}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$

Step 6

Multiply $- \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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

$\frac{\sqrt{2}}{4} - \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2}$

Combine using the product rule for radicals.

$\frac{\sqrt{2}}{4} - \frac{\sqrt{2 \cdot 3}}{2 \cdot 2}$

Multiply $2$ by $3$ .

$\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$

Decimal Form:

$- 0.25881904 \dots$