Find the Exact Value csc(75 degrees )

$\csc (75 °)$

Step 1

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

$\csc (30 + 45)$

Step 2

Apply the sum of angles identity.

$\frac{\sec (30) \sec (45) \csc (30) \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)}$

Step 3

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

$\frac{\frac{2}{\sqrt{3}} \sec (45) \csc (30) \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)}$

Step 4

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

$\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \csc (30) \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)}$

Step 5

The exact value of $\csc (30)$ is $2$ .

$\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \csc (45)}{\sec (30) \csc (45) + \csc (30) \sec (45)}$

Step 6

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

$\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\sec (30) \csc (45) + \csc (30) \sec (45)}$

Step 7

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

$\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \csc (45) + \csc (30) \sec (45)}$

Step 8

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

$\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + \csc (30) \sec (45)}$

Step 9

The exact value of $\csc (30)$ is $2$ .

$\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \sec (45)}$

Step 10

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

$\frac{\frac{2}{\sqrt{3}} \cdot \frac{2}{\sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Step 11

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

Simplify the numerator.

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

$\frac{\frac{2 \cdot 2}{\sqrt{3} \sqrt{2}} \cdot 2 \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Combine $\frac{2 \cdot 2}{\sqrt{3} \sqrt{2}}$ and $2$ .

$\frac{\frac{2 \cdot 2 \cdot 2}{\sqrt{3} \sqrt{2}} \sqrt{2}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Combine $\frac{2 \cdot 2 \cdot 2}{\sqrt{3} \sqrt{2}}$ and $\sqrt{2}$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Simplify the denominator.

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Combine and simplify the denominator.

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Add $1$ and $1$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Rewrite the expression.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3^{1}} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

Evaluate the exponent.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3}}{3} \sqrt{2} + 2 \frac{2}{\sqrt{2}}}$

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

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{3} \sqrt{2}}{3} + 2 \frac{2}{\sqrt{2}}}$

Combine using the product rule for radicals.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{2 \cdot 3}}{3} + 2 \frac{2}{\sqrt{2}}}$

Multiply $2$ by $3$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2}{\sqrt{2}}}$

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

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

Combine and simplify the denominator.

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}}$

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}}$

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

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

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

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

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

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}}$

Rewrite the expression.

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

Evaluate the exponent.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \frac{2 \sqrt{2}}{2}}$

Rewrite the expression.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \sqrt{2}}$

To write $2 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + 2 \sqrt{2} \cdot \frac{3}{3}}$

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

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6}}{3} + \frac{2 \sqrt{2} \cdot 3}{3}}$

Combine the numerators over the common denominator.

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6} + 2 \sqrt{2} \cdot 3}{3}}$

Multiply $3$ by $2$ .

$\frac{\frac{2 \cdot 2 \cdot 2 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Simplify the numerator.

Multiply $2$ by $2$ .

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

Multiply $4$ by $2$ .

$\frac{\frac{8 \sqrt{2}}{\sqrt{3} \sqrt{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Simplify the denominator.

Combine using the product rule for radicals.

$\frac{\frac{8 \sqrt{2}}{\sqrt{3 \cdot 2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Multiply $3$ by $2$ .

$\frac{\frac{8 \sqrt{2}}{\sqrt{6}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Simplify the numerator.

Combine $\sqrt{2}$ and $\sqrt{6}$ into a single radical.

$\frac{8 \sqrt{\frac{2}{6}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$\frac{8 \sqrt{\frac{2 (1)}{6}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Cancel the common factor.

$\frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Rewrite the expression.

$\frac{8 \sqrt{\frac{1}{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\frac{8 \frac{\sqrt{1}}{\sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Any root of $1$ is $1$ .

$\frac{8 \frac{1}{\sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{8 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{8 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

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

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Add $1$ and $1$ .

$\frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{8 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

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

$\frac{8 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

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

$\frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Rewrite the expression.

$\frac{8 \frac{\sqrt{3}}{3^{1}}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Evaluate the exponent.

$\frac{8 \frac{\sqrt{3}}{3}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Combine $8$ and $\frac{\sqrt{3}}{3}$ .

$\frac{\frac{8 \sqrt{3}}{3}}{\frac{2 \sqrt{6} + 6 \sqrt{2}}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 \sqrt{3}}{3} \cdot \frac{3}{2 \sqrt{6} + 6 \sqrt{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{8 \sqrt{3}}{3} \cdot \frac{3}{2 \sqrt{6} + 6 \sqrt{2}}$

Rewrite the expression.

$8 \sqrt{3} \frac{1}{2 \sqrt{6} + 6 \sqrt{2}}$

Combine $\frac{1}{2 \sqrt{6} + 6 \sqrt{2}}$ and $8$ .

$\frac{8}{2 \sqrt{6} + 6 \sqrt{2}} \sqrt{3}$

Combine $\frac{8}{2 \sqrt{6} + 6 \sqrt{2}}$ and $\sqrt{3}$ .

$\frac{8 \sqrt{3}}{2 \sqrt{6} + 6 \sqrt{2}}$

Cancel the common factor of $8$ and $2 \sqrt{6} + 6 \sqrt{2}$ .

Factor $2$ out of $8 \sqrt{3}$ .

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

Cancel the common factors.

Factor $2$ out of $2 \sqrt{6}$ .

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

Factor $2$ out of $6 \sqrt{2}$ .

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

Factor $2$ out of $2 (\sqrt{6}) + 2 (3 \sqrt{2})$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{4 \sqrt{3}}{\sqrt{6} + 3 \sqrt{2}}$

Multiply $\frac{4 \sqrt{3}}{\sqrt{6} + 3 \sqrt{2}}$ by $\frac{\sqrt{6} - 3 \sqrt{2}}{\sqrt{6} - 3 \sqrt{2}}$ .

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

Multiply $\frac{4 \sqrt{3}}{\sqrt{6} + 3 \sqrt{2}}$ by $\frac{\sqrt{6} - 3 \sqrt{2}}{\sqrt{6} - 3 \sqrt{2}}$ .

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

Expand the denominator using the FOIL method.

$\frac{4 \sqrt{3} (\sqrt{6} - 3 \sqrt{2})}{{\sqrt{6}}^{2} - 3 \sqrt{12} + 3 \sqrt{12} - 9 {\sqrt{2}}^{2}}$

Simplify.

$\frac{4 \sqrt{3} (\sqrt{6} - 3 \sqrt{2})}{- 12}$

Cancel the common factor of $4$ and $- 12$ .

Factor $4$ out of $4 \sqrt{3} (\sqrt{6} - 3 \sqrt{2})$ .

$\frac{4 (\sqrt{3} (\sqrt{6} - 3 \sqrt{2}))}{- 12}$

Cancel the common factors.

Factor $4$ out of $- 12$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{3} (\sqrt{6} - 3 \sqrt{2})}{- 3}$

Apply the distributive property.

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

Combine using the product rule for radicals.

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

Multiply $\sqrt{3} (- 3 \sqrt{2})$ .

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 6} - 3 \sqrt{3 \cdot 2}}{- 3}$

Multiply $3$ by $2$ .

$\frac{\sqrt{3 \cdot 6} - 3 \sqrt{6}}{- 3}$

Simplify each term.

Multiply $3$ by $6$ .

$\frac{\sqrt{18} - 3 \sqrt{6}}{- 3}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)} - 3 \sqrt{6}}{- 3}$

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 2} - 3 \sqrt{6}}{- 3}$

Pull terms out from under the radical.

$\frac{3 \sqrt{2} - 3 \sqrt{6}}{- 3}$

Cancel the common factor of $3 \sqrt{2} - 3 \sqrt{6}$ and $- 3$ .

Factor $3$ out of $3 \sqrt{2}$ .

$\frac{3 (\sqrt{2}) - 3 \sqrt{6}}{- 3}$

Factor $3$ out of $- 3 \sqrt{6}$ .

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

Factor $3$ out of $3 (\sqrt{2}) + 3 (- \sqrt{6})$ .

$\frac{3 (\sqrt{2} - \sqrt{6})}{- 3}$

Move the negative one from the denominator of $\frac{\sqrt{2} - \sqrt{6}}{- 1}$ .

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

Rewrite $- 1 \cdot (\sqrt{2} - \sqrt{6})$ as $- (\sqrt{2} - \sqrt{6})$ .

$- (\sqrt{2} - \sqrt{6})$

Apply the distributive property.

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

Multiply $- - \sqrt{6}$ .

Multiply $- 1$ by $- 1$ .

$- \sqrt{2} + 1 \sqrt{6}$

Multiply $\sqrt{6}$ by $1$ .

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.03527618 \dots$

Find the Exact Value csc(75 degrees )

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