Find the Exact Value cos(arccsc((2)^-1))

$\cos (arccsc ({(2)}^{- 1}))$

Draw a triangle in the plane with vertices $(\sqrt{{({(2)}^{- 1})}^{2} - 1^{2}}, 1)$ , $(\sqrt{{({(2)}^{- 1})}^{2} - 1^{2}}, 0)$ , and the origin. Then $arccsc ({(2)}^{- 1})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{{({(2)}^{- 1})}^{2} - 1^{2}}, 1)$ . Therefore, $\cos (arccsc ({(2)}^{- 1}))$ is $\sqrt{{(2^{- 1})}^{2} - 1} \cdot 2$ .

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

Multiply the exponents in ${(2^{- 1})}^{2}$ .

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

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

Multiply $- 1$ by $2$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Raise $2$ to the power of $2$ .

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $- 1$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $4$ .

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

Subtract $4$ from $1$ .

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

Move the negative in front of the fraction.

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

Rewrite $- \frac{3}{4}$ as ${(\frac{i}{2})}^{2} \cdot 3$ .

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{3}{4}} \cdot 2$

Factor the perfect power $1^{2}$ out of $3$ .

$\sqrt{i^{2} \frac{1^{2} \cdot 3}{4}} \cdot 2$

Factor the perfect power $2^{2}$ out of $4$ .

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

Rearrange the fraction $\frac{1^{2} \cdot 3}{2^{2} \cdot 1}$ .

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

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

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

Pull terms out from under the radical.

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$i \sqrt{3}$

Find the Exact Value cos(arccsc((2)^-1))

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