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

$\cos (75 °) \cos (15 °) - \sin (75 °) \sin (15 °)$

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.

$\cos (30 + 45) \cos (15 °) - \sin (75 °) \sin (15 °)$

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

$(\cos (30) \cos (45) - \sin (30) \sin (45)) \cos (15 °) - \sin (75 °) \sin (15 °)$

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

$(\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)) \cos (15 °) - \sin (75 °) \sin (15 °)$

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

$(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)) \cos (15 °) - \sin (75 °) \sin (15 °)$

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

$(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)) \cos (15 °) - \sin (75 °) \sin (15 °)$

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

$(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15 °) - \sin (75 °) \sin (15 °)$

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

$(\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15 °) - \sin (75 °) \sin (15 °)$

Combine using the product rule for radicals.

$(\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15 °) - \sin (75 °) \sin (15 °)$

Multiply $3$ by $2$ .

$(\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15 °) - \sin (75 °) \sin (15 °)$

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) \cos (15 °) - \sin (75 °) \sin (15 °)$

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

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

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cos (15 °) - \sin (75 °) \sin (15 °)$

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cos (15 °) - \sin (75 °) \sin (15 °)$

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cos (15 °) - \sin (75 °) \sin (15 °)$

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.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cos (45 - 30) - \sin (75 °) \sin (15 °)$

Separate negation.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cos (45 - (30)) - \sin (75 °) \sin (15 °)$

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

$\frac{\sqrt{6} - \sqrt{2}}{4} (\cos (45) \cos (30) + \sin (45) \sin (30)) - \sin (75 °) \sin (15 °)$

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

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) - \sin (75 °) \sin (15 °)$

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

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) - \sin (75 °) \sin (15 °)$

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

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) - \sin (75 °) \sin (15 °)$

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

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (75 °) \sin (15 °)$

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

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (75 °) \sin (15 °)$

Combine using the product rule for radicals.

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (75 °) \sin (15 °)$

Multiply $2$ by $3$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (75 °) \sin (15 °)$

Multiply $2$ by $2$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) - \sin (75 °) \sin (15 °)$

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

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

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) - \sin (75 °) \sin (15 °)$

Multiply $2$ by $2$ .

$\frac{\sqrt{6} - \sqrt{2}}{4} (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) - \sin (75 °) \sin (15 °)$

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4} - \sin (75 °) \sin (15 °)$

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{6} + \sqrt{2}}{4}$ .

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

$\frac{(\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}{4 \cdot 4} - \sin (75 °) \sin (15 °)$

Multiply $4$ by $4$ .

$\frac{(\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})}{16} - \sin (75 °) \sin (15 °)$

Simplify the numerator.

Expand $(\sqrt{6} - \sqrt{2}) (\sqrt{6} + \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{6} (\sqrt{6} + \sqrt{2}) - \sqrt{2} (\sqrt{6} + \sqrt{2})}{16} - \sin (75 °) \sin (15 °)$

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2} - \sqrt{2} (\sqrt{6} + \sqrt{2})}{16} - \sin (75 °) \sin (15 °)$

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{6} + \sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 6} + \sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Multiply $6$ by $6$ .

$\frac{\sqrt{36} + \sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{6^{2}} + \sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{6 + \sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Combine using the product rule for radicals.

$\frac{6 + \sqrt{6 \cdot 2} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Multiply $6$ by $2$ .

$\frac{6 + \sqrt{12} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{6 + \sqrt{4 (3)} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Rewrite $4$ as $2^{2}$ .

$\frac{6 + \sqrt{2^{2} \cdot 3} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Pull terms out from under the radical.

$\frac{6 + 2 \sqrt{3} - \sqrt{2} \sqrt{6} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Multiply $- \sqrt{2} \sqrt{6}$ .

Combine using the product rule for radicals.

$\frac{6 + 2 \sqrt{3} - \sqrt{6 \cdot 2} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Multiply $6$ by $2$ .

$\frac{6 + 2 \sqrt{3} - \sqrt{12} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{6 + 2 \sqrt{3} - \sqrt{4 (3)} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Rewrite $4$ as $2^{2}$ .

$\frac{6 + 2 \sqrt{3} - \sqrt{2^{2} \cdot 3} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Pull terms out from under the radical.

$\frac{6 + 2 \sqrt{3} - (2 \sqrt{3}) - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Multiply $2$ by $- 1$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{2} \sqrt{2}}{16} - \sin (75 °) \sin (15 °)$

Multiply $- \sqrt{2} \sqrt{2}$ .

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - ({\sqrt{2}}^{1} \sqrt{2})}{16} - \sin (75 °) \sin (15 °)$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{16} - \sin (75 °) \sin (15 °)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{2}}^{1 + 1}}{16} - \sin (75 °) \sin (15 °)$

Add $1$ and $1$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - {\sqrt{2}}^{2}}{16} - \sin (75 °) \sin (15 °)$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{16} - \sin (75 °) \sin (15 °)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{16} - \sin (75 °) \sin (15 °)$

Combine $\frac{1}{2}$ and $2$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{\frac{2}{2}}}{16} - \sin (75 °) \sin (15 °)$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{\frac{2}{2}}}{16} - \sin (75 °) \sin (15 °)$

Rewrite the expression.

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2^{1}}{16} - \sin (75 °) \sin (15 °)$

Evaluate the exponent.

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 1 \cdot 2}{16} - \sin (75 °) \sin (15 °)$

Multiply $- 1$ by $2$ .

$\frac{6 + 2 \sqrt{3} - 2 \sqrt{3} - 2}{16} - \sin (75 °) \sin (15 °)$

Subtract $2$ from $6$ .

$\frac{4 + 2 \sqrt{3} - 2 \sqrt{3}}{16} - \sin (75 °) \sin (15 °)$

Subtract $2 \sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{4 + 0}{16} - \sin (75 °) \sin (15 °)$

Add $4$ and $0$ .

$\frac{4}{16} - \sin (75 °) \sin (15 °)$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$\frac{4 (1)}{16} - \sin (75 °) \sin (15 °)$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{4 \cdot 1}{4 \cdot 4} - \sin (75 °) \sin (15 °)$

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 4} - \sin (75 °) \sin (15 °)$

Rewrite the expression.

$\frac{1}{4} - \sin (75 °) \sin (15 °)$

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

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

$\frac{1}{4} - \sin (30 + 45) \sin (15 °)$

Apply the sum of angles identity.

$\frac{1}{4} - (\sin (30) \cos (45) + \cos (30) \sin (45)) \sin (15 °)$

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

$\frac{1}{4} - (\frac{1}{2} \cos (45) + \cos (30) \sin (45)) \sin (15 °)$

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

$\frac{1}{4} - (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \sin (15 °)$

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

$\frac{1}{4} - (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) \sin (15 °)$

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

$\frac{1}{4} - (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (15 °)$

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

Simplify each term.

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

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

$\frac{1}{4} - (\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (15 °)$

Multiply $2$ by $2$ .

$\frac{1}{4} - (\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \sin (15 °)$

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

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

$\frac{1}{4} - (\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \sin (15 °)$

Combine using the product rule for radicals.

$\frac{1}{4} - (\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \sin (15 °)$

Multiply $3$ by $2$ .

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

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.

$\frac{1}{4} - \frac{\sqrt{2} + \sqrt{6}}{4} \sin (45 - 30)$

Separate negation.

$\frac{1}{4} - \frac{\sqrt{2} + \sqrt{6}}{4} \sin (45 - (30))$

Apply the difference of angles identity.

$\frac{1}{4} - \frac{\sqrt{2} + \sqrt{6}}{4} (\sin (45) \cos (30) - \cos (45) \sin (30))$

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

$\frac{1}{4} - \frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

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

$\frac{1}{4} - \frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

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

$\frac{1}{4} - \frac{\sqrt{2} + \sqrt{6}}{4} (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Multiply $2$ by $2$ .

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

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

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

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

Multiply $2$ by $2$ .

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

Combine the numerators over the common denominator.

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

Multiply $- \frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{\sqrt{6} - \sqrt{2}}{4}$ .

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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

Multiply $4$ by $4$ .

$\frac{1}{4} - \frac{(\sqrt{6} - \sqrt{2}) (\sqrt{2} + \sqrt{6})}{16}$

Simplify the numerator.

Expand $(\sqrt{6} - \sqrt{2}) (\sqrt{2} + \sqrt{6})$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{4} - \frac{\sqrt{6} (\sqrt{2} + \sqrt{6}) - \sqrt{2} (\sqrt{2} + \sqrt{6})}{16}$

Apply the distributive property.

$\frac{1}{4} - \frac{\sqrt{6} \sqrt{2} + \sqrt{6} \sqrt{6} - \sqrt{2} (\sqrt{2} + \sqrt{6})}{16}$

Apply the distributive property.

$\frac{1}{4} - \frac{\sqrt{6} \sqrt{2} + \sqrt{6} \sqrt{6} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{1}{4} - \frac{\sqrt{6 \cdot 2} + \sqrt{6} \sqrt{6} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Multiply $6$ by $2$ .

$\frac{1}{4} - \frac{\sqrt{12} + \sqrt{6} \sqrt{6} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{1}{4} - \frac{\sqrt{4 (3)} + \sqrt{6} \sqrt{6} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Rewrite $4$ as $2^{2}$ .

$\frac{1}{4} - \frac{\sqrt{2^{2} \cdot 3} + \sqrt{6} \sqrt{6} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Pull terms out from under the radical.

$\frac{1}{4} - \frac{2 \sqrt{3} + \sqrt{6} \sqrt{6} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Combine using the product rule for radicals.

$\frac{1}{4} - \frac{2 \sqrt{3} + \sqrt{6 \cdot 6} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Multiply $6$ by $6$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + \sqrt{36} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Rewrite $36$ as $6^{2}$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + \sqrt{6^{2}} - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - \sqrt{2} \sqrt{2} - \sqrt{2} \sqrt{6}}{16}$

Multiply $- \sqrt{2} \sqrt{2}$ .

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - ({\sqrt{2}}^{1} \sqrt{2}) - \sqrt{2} \sqrt{6}}{16}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) - \sqrt{2} \sqrt{6}}{16}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - {\sqrt{2}}^{1 + 1} - \sqrt{2} \sqrt{6}}{16}$

Add $1$ and $1$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - {\sqrt{2}}^{2} - \sqrt{2} \sqrt{6}}{16}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - {(2^{\frac{1}{2}})}^{2} - \sqrt{2} \sqrt{6}}{16}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2^{\frac{1}{2} \cdot 2} - \sqrt{2} \sqrt{6}}{16}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2^{\frac{2}{2}} - \sqrt{2} \sqrt{6}}{16}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2^{\frac{2}{2}} - \sqrt{2} \sqrt{6}}{16}$

Rewrite the expression.

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2^{1} - \sqrt{2} \sqrt{6}}{16}$

Evaluate the exponent.

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 1 \cdot 2 - \sqrt{2} \sqrt{6}}{16}$

Multiply $- 1$ by $2$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2 - \sqrt{2} \sqrt{6}}{16}$

Multiply $- \sqrt{2} \sqrt{6}$ .

Combine using the product rule for radicals.

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2 - \sqrt{6 \cdot 2}}{16}$

Multiply $6$ by $2$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2 - \sqrt{12}}{16}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2 - \sqrt{4 (3)}}{16}$

Rewrite $4$ as $2^{2}$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2 - \sqrt{2^{2} \cdot 3}}{16}$

Pull terms out from under the radical.

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2 - (2 \sqrt{3})}{16}$

Multiply $2$ by $- 1$ .

$\frac{1}{4} - \frac{2 \sqrt{3} + 6 - 2 - 2 \sqrt{3}}{16}$

Subtract $2 \sqrt{3}$ from $2 \sqrt{3}$ .

$\frac{1}{4} - \frac{0 + 6 - 2}{16}$

Add $0$ and $6$ .

$\frac{1}{4} - \frac{6 - 2}{16}$

Subtract $2$ from $6$ .

$\frac{1}{4} - \frac{4}{16}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$\frac{1}{4} - \frac{4 (1)}{16}$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{1}{4} - \frac{4 \cdot 1}{4 \cdot 4}$

Cancel the common factor.

$\frac{1}{4} - \frac{4 \cdot 1}{4 \cdot 4}$

Rewrite the expression.

$\frac{1}{4} - \frac{1}{4}$

Step 2

Combine fractions.

Combine the numerators over the common denominator.

$\frac{1 - 1}{4}$

Simplify the expression.

Subtract $1$ from $1$ .

$\frac{0}{4}$

Divide $0$ by $4$ .

$0$

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

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