Find the Exact Value sec(750 degrees )-csc(-300 degrees )

$\sec (750 °) - \csc (- 300 °)$

Simplify each term.

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sec (30) - \csc (- 300 °)$

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

$\frac{2}{\sqrt{3}} - \csc (- 300 °)$

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

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - \csc (- 300 °)$

Combine and simplify the denominator.

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

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} - \csc (- 300 °)$

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \csc (- 300 °)$

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \csc (- 300 °)$

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \csc (- 300 °)$

Add $1$ and $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} - \csc (- 300 °)$

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

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

$\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \csc (- 300 °)$

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

$\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \csc (- 300 °)$

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

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \csc (- 300 °)$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \csc (- 300 °)$

Rewrite the expression.

$\frac{2 \sqrt{3}}{3^{1}} - \csc (- 300 °)$

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3} - \csc (- 300 °)$

The exact value of $\csc (- 300 °)$ is $\frac{2 \sqrt{3}}{3}$ .

Rewrite $- 300 °$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{2 \sqrt{3}}{3} - \csc (\frac{- 600}{2})$

Apply the reciprocal identity to $\csc (\frac{- 600}{2})$ .

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sin (\frac{- 600}{2})}$

Apply the sine half-angle identity.

$\frac{2 \sqrt{3}}{3} - \frac{1}{\pm \sqrt{\frac{1 - \cos (- 600)}{2}}}$

Change the $\pm$ to $+$ because cosecant is positive in the first quadrant.

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{1 - \cos (- 600)}{2}}}$

Simplify $\frac{1}{\sqrt{\frac{1 - \cos (- 600)}{2}}}$ .

Simplify the numerator.

Add full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{1 - \cos (120)}{2}}}$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{1 - - \cos (60)}{2}}}$

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

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{1 - - \frac{1}{2}}{2}}}$

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

Multiply $- 1$ by $- 1$ .

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

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

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{1 + \frac{1}{2}}{2}}}$

Write $1$ as a fraction with a common denominator.

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{1}{2}}{2}}}$

Combine the numerators over the common denominator.

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{\frac{2 + 1}{2}}{2}}}$

Add $2$ and $1$ .

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{\frac{3}{2}}{2}}}$

Simplify the denominator.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

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

Multiply $2$ by $2$ .

$\frac{2 \sqrt{3}}{3} - \frac{1}{\sqrt{\frac{3}{4}}}$

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\frac{2 \sqrt{3}}{3} - \frac{1}{\frac{\sqrt{3}}{\sqrt{4}}}$

Simplify the denominator.

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

$\frac{2 \sqrt{3}}{3} - \frac{1}{\frac{\sqrt{3}}{\sqrt{2^{2}}}}$

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

$\frac{2 \sqrt{3}}{3} - \frac{1}{\frac{\sqrt{3}}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 \sqrt{3}}{3} - (1 \frac{2}{\sqrt{3}})$

Multiply $\frac{2}{\sqrt{3}}$ by $1$ .

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

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

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

Combine and simplify the denominator.

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

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

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

$\frac{2 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

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

$\frac{2 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3^{1}}$

Evaluate the exponent.

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

Simplify terms.

Combine the numerators over the common denominator.

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

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

$\frac{0}{3}$

Divide $0$ by $3$ .

$0$

Find the Exact Value sec(750 degrees )-csc(-300 degrees )

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