Find the Exact Value cos(25 degrees )cos(15 degrees )-sin(25 degrees )sin(15 degrees )

$\cos (25 °) \cos (15 °) - \sin (25 °) \sin (15 °)$

Step 1

Simplify terms.

Simplify each term.

The exact value of $\cos (15 °)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\cos (25 °) \cos (45 - 30) - \sin (25 °) \sin (15 °)$

Separate negation.

$\cos (25 °) \cos (45 - (30)) - \sin (25 °) \sin (15 °)$

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

$\cos (25 °) (\cos (45) \cos (30) + \sin (45) \sin (30)) - \sin (25 °) \sin (15 °)$

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

$\cos (25 °) (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) - \sin (25 °) \sin (15 °)$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\cos (25 °) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) - \sin (25 °) \sin (15 °)$

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

$\cos (25 °) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) - \sin (25 °) \sin (15 °)$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\cos (25 °) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25 °) \sin (15 °)$

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Simplify each term.

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

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

$\cos (25 °) (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25 °) \sin (15 °)$

Combine using the product rule for radicals.

$\cos (25 °) (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25 °) \sin (15 °)$

Multiply $2$ by $3$ .

$\cos (25 °) (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25 °) \sin (15 °)$

Multiply $2$ by $2$ .

$\cos (25 °) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (25 °) \sin (15 °)$

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

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

$\cos (25 °) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) - \sin (25 °) \sin (15 °)$

Multiply $2$ by $2$ .

$\cos (25 °) (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) - \sin (25 °) \sin (15 °)$

Combine the numerators over the common denominator.

$\cos (25 °) \frac{\sqrt{6} + \sqrt{2}}{4} - \sin (25 °) \sin (15 °)$

Combine $\cos (25 °)$ and $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) \sin (15 °)$

The exact value of $\sin (15 °)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) \sin (45 - 30)$

Separate negation.

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) \sin (45 - (30))$

Apply the difference of angles identity.

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\sin (45) \cos (30) - \cos (45) \sin (30))$

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

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

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

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Simplify each term.

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

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

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Combine using the product rule for radicals.

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $3$ .

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $2$ .

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

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

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

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Multiply $2$ by $2$ .

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Combine the numerators over the common denominator.

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \sin (25 °) \frac{\sqrt{6} - \sqrt{2}}{4}$

Combine $\frac{\sqrt{6} - \sqrt{2}}{4}$ and $\sin (25 °)$ .

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2})}{4} - \frac{(\sqrt{6} - \sqrt{2}) \sin (25 °)}{4}$

Combine the numerators over the common denominator.

$\frac{\cos (25 °) (\sqrt{6} + \sqrt{2}) - (\sqrt{6} - \sqrt{2}) \sin (25 °)}{4}$

Step 2

Simplify the numerator.

Apply the distributive property.

$\frac{\cos (25 °) \sqrt{6} + \cos (25 °) \sqrt{2} - (\sqrt{6} - \sqrt{2}) \sin (25 °)}{4}$

Apply the distributive property.

$\frac{\cos (25 °) \sqrt{6} + \cos (25 °) \sqrt{2} + (- \sqrt{6} - - \sqrt{2}) \sin (25 °)}{4}$

Multiply $- - \sqrt{2}$ .

Multiply $- 1$ by $- 1$ .

$\frac{\cos (25 °) \sqrt{6} + \cos (25 °) \sqrt{2} + (- \sqrt{6} + 1 \sqrt{2}) \sin (25 °)}{4}$

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

$\frac{\cos (25 °) \sqrt{6} + \cos (25 °) \sqrt{2} + (- \sqrt{6} + \sqrt{2}) \sin (25 °)}{4}$

Apply the distributive property.

$\frac{\cos (25 °) \sqrt{6} + \cos (25 °) \sqrt{2} - \sqrt{6} \sin (25 °) + \sqrt{2} \sin (25 °)}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\cos (25 °) \sqrt{6} + \cos (25 °) \sqrt{2} - \sqrt{6} \sin (25 °) + \sqrt{2} \sin (25 °)}{4}$

Decimal Form:

$0.76604444 \dots$

Find the Exact Value cos(25 degrees )cos(15 degrees )-sin(25 degrees )sin(15 degrees )

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