Verify the Identity csc(x) to the power of four minus csc(x) to the power of two equals cot(x) to the power of four plus cot(x) to the power of two

$\csc^{4} (x) - \csc^{2} (x) = \cot^{4} (x) + \cot^{2} (x)$

Start on the right side.

$\cot^{4} (x) + \cot^{2} (x)$

Factor $\cot^{2} (x)$ out of $\cot^{4} (x) + \cot^{2} (x)$ .

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Factor $\cot^{2} (x)$ out of $\cot^{4} (x)$ .

$\cot^{2} (x) \cot^{2} (x) + \cot^{2} (x)$

First, multiply by $1$ .

$\cot^{2} (x) \cot^{2} (x) + \cot^{2} (x) \cdot 1$

Factor $\cot^{2} (x)$ out of $\cot^{2} (x) \cot^{2} (x) + \cot^{2} (x) \cdot 1$ .

$\cot^{2} (x) (\cot^{2} (x) + 1)$

Apply Pythagorean identity.

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Should be rewritten terms.

$\cot^{2} (x) (1 + \cot^{2} (x))$

Apply pythagorean identity.

$\cot^{2} (x) \csc^{2} (x)$

Apply Pythagorean identity in reverse.

$(\csc^{2} (x) - 1) \csc^{2} (x)$

Convert to sines and cosines.

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Apply the reciprocal identity to $\csc (x)$ .

$({(\frac{1}{\sin (x)})}^{2} - 1) \csc^{2} (x)$

Apply the reciprocal identity to $\csc (x)$ .

$({(\frac{1}{\sin (x)})}^{2} - 1) {(\frac{1}{\sin (x)})}^{2}$

Apply the product rule to $\frac{1}{\sin (x)}$ .

$(\frac{1^{2}}{\sin^{2} (x)} - 1) {(\frac{1}{\sin (x)})}^{2}$

Apply the product rule to $\frac{1}{\sin (x)}$ .

$(\frac{1^{2}}{\sin^{2} (x)} - 1) \frac{1^{2}}{\sin^{2} (x)}$

Simplify.

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One raised to any power equals one.

$(\frac{1^{2}}{{\sin (x)}^{2}} - 1) \frac{1}{{\sin (x)}^{2}}$

One to any power is one.

$(\frac{1}{{\sin (x)}^{2}} - 1) \frac{1}{{\sin (x)}^{2}}$

Use the distributive law.

$\frac{1}{{\sin (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}} - 1 \frac{1}{{\sin (x)}^{2}}$

Multiply $\frac{1}{{\sin (x)}^{2}} \cdot \frac{1}{{\sin (x)}^{2}}$ .

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Multiply $\frac{1}{{\sin (x)}^{2}}$ and $\frac{1}{{\sin (x)}^{2}}$ .

$\frac{1}{{\sin (x)}^{2} {\sin (x)}^{2}} - 1 \frac{1}{{\sin (x)}^{2}}$

Multiply ${\sin (x)}^{2}$ by ${\sin (x)}^{2}$ by adding the exponents.

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Apply the power rule $a^{m} a^{n} = a^{m + n}$ to Add exponents.

$\frac{1}{{\sin (x)}^{2 + 2}} - 1 \frac{1}{{\sin (x)}^{2}}$

Add $2$ and $2$ .

$\frac{1}{{\sin (x)}^{4}} - 1 \frac{1}{{\sin (x)}^{2}}$

Rewrite $- 1 \frac{1}{{\sin (x)}^{2}}$ as $- \frac{1}{{\sin (x)}^{2}}$ .

$\frac{1}{{\sin (x)}^{4}} - \frac{1}{{\sin (x)}^{2}}$

To write $- \frac{1}{{\sin (x)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$ .

$\frac{1}{{\sin (x)}^{4}} - \frac{1}{{\sin (x)}^{2}} \cdot \frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$

Write each expression with a common denominator of ${\sin (x)}^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Add.

$\frac{1}{{\sin (x)}^{4}} - \frac{1 {\sin (x)}^{2}}{{\sin (x)}^{2} {\sin (x)}^{2}}$

Multiply ${\sin (x)}^{2}$ by ${\sin (x)}^{2}$ by adding the exponents.

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

$\frac{1}{{\sin (x)}^{4}} - \frac{1 {\sin (x)}^{2}}{{\sin (x)}^{2 + 2}}$

Add $2$ and $2$ .

$\frac{1}{{\sin (x)}^{4}} - \frac{1 {\sin (x)}^{2}}{{\sin (x)}^{4}}$

Add numerators over the common denominator.

$\frac{1 - {\sin (x)}^{2}}{{\sin (x)}^{4}}$

Clarify the numerator.

$\frac{(1 + \sin (x)) (1 - \sin (x))}{\sin^{4} (x)}$

Now consider the left side of the equation.

$\csc^{4} (x) - \csc^{2} (x)$

Convert to sines and cosines.

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Apply the reciprocal identity to $\csc (x)$ .

${(\frac{1}{\sin (x)})}^{4} - \csc^{2} (x)$

Apply the reciprocal identity to $\csc (x)$ .

${(\frac{1}{\sin (x)})}^{4} - {(\frac{1}{\sin (x)})}^{2}$

Apply the product rule to $\frac{1}{\sin (x)}$ .

$\frac{1^{4}}{\sin^{4} (x)} - {(\frac{1}{\sin (x)})}^{2}$

Apply the product rule to $\frac{1}{\sin (x)}$ .

$\frac{1^{4}}{\sin^{4} (x)} - \frac{1^{2}}{\sin^{2} (x)}$

Simplify.

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Clarify each term.

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One to any power is one.

$\frac{1}{{\sin (x)}^{4}} - \frac{1^{2}}{{\sin (x)}^{2}}$

One to any power is one.

$\frac{1}{{\sin (x)}^{4}} - \frac{1}{{\sin (x)}^{2}}$

To write $- \frac{1}{{\sin (x)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$ .

$\frac{1}{{\sin (x)}^{4}} - \frac{1}{{\sin (x)}^{2}} \cdot \frac{{\sin (x)}^{2}}{{\sin (x)}^{2}}$

Write each expression with a common denominator of ${\sin (x)}^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{1}{{\sin (x)}^{4}} - \frac{1 {\sin (x)}^{2}}{{\sin (x)}^{2} {\sin (x)}^{2}}$

Multiply ${\sin (x)}^{2}$ by ${\sin (x)}^{2}$ by adding the exponents.

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

$\frac{1}{{\sin (x)}^{4}} - \frac{1 {\sin (x)}^{2}}{{\sin (x)}^{2 + 2}}$

Add $2$ and $2$ .

$\frac{1}{{\sin (x)}^{4}} - \frac{1 {\sin (x)}^{2}}{{\sin (x)}^{4}}$

Combine the numerators over the common denominator.

$\frac{1 - {\sin (x)}^{2}}{{\sin (x)}^{4}}$

Clarify the numerator.

$\frac{(1 + \sin (x)) (1 - \sin (x))}{\sin^{4} (x)}$

Because the two sides have been shown to be equivalent, the equation is an identity.

$\csc^{4} (x) - \csc^{2} (x) = \cot^{4} (x) + \cot^{2} (x)$ is an identity

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