Find the Exact Value csc((7pi)/4)^2

$\csc^{2} (\frac{7 π}{4})$

Step 1

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

${(- \csc (\frac{π}{4}))}^{2}$

Step 2

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

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

Step 3

Simplify the expression.

Apply the product rule to $- \sqrt{2}$ .

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

Raise $- 1$ to the power of $2$ .

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

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

${\sqrt{2}}^{2}$

Step 4

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

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

${(2^{\frac{1}{2}})}^{2}$

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

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

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

$2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2^{\frac{2}{2}}$

Rewrite the expression.

$2^{1}$

Evaluate the exponent.

$2$

Find the Exact Value csc((7pi)/4)^2

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