Find the Exact Value cos(75 degrees )-cos(15 degrees )

$\cos (75 °) - \cos (15 °)$

Step 1

Simplify each term.

The exact value of $\cos (75 °)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\cos (30 + 45) - \cos (15 °)$

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

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

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

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

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

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

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

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45) - \cos (15 °)$

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

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

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

Simplify each term.

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

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.

$\frac{\sqrt{6} - \sqrt{2}}{4} - \cos (45 - 30)$

Separate negation.

$\frac{\sqrt{6} - \sqrt{2}}{4} - \cos (45 - (30))$

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

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} - (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $2$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} - (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

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

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

$\frac{\sqrt{6} - \sqrt{2}}{4} - (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2})$

Multiply $2$ by $2$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} - (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4})$

Combine the numerators over the common denominator.

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

Step 2

Simplify terms.

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2} - (\sqrt{6} + \sqrt{2})}{4}$

Apply the distributive property.

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

Subtract $\sqrt{6}$ from $\sqrt{6}$ .

$\frac{0 - \sqrt{2} - \sqrt{2}}{4}$

Subtract $\sqrt{2}$ from $0$ .

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

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

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

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2 \sqrt{2}$ .

$\frac{2 (- \sqrt{2})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 (- \sqrt{2})}{2 (2)}$

Cancel the common factor.

$\frac{2 (- \sqrt{2})}{2 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.70710678 \dots$

Find the Exact Value cos(75 degrees )-cos(15 degrees )

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