Verify the Identity csc(x)^4-csc(x)^2=cot(x)^4+cot(x)^2

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Trigonometry

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

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 $\cot^{4} (x)$ .

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

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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Rearrange 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 to any power is one.

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

One to any power is one.

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

Apply the distributive property.

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

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

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

Simplify 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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Simplify 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.

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

Simplify 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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Name

Name	four hundred ninety-nine million five hundred twenty-seven thousand two hundred forty-five

Interesting facts

499527245 has 32 divisors, whose sum is 699217920
The reverse of 499527245 is 542725994
Previous prime number is 179

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 47
Digital Root 2

Name

Name	five hundred seventy-four million four hundred twenty-six thousand four hundred thirty-two

Interesting facts

574426432 has 512 divisors, whose sum is 6811630200
The reverse of 574426432 is 234624475
Previous prime number is 173

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 37
Digital Root 1

Name

Name	one billion five hundred nine million five hundred eleven thousand three hundred forty-six

Interesting facts

1509511346 has 16 divisors, whose sum is 2957410176
The reverse of 1509511346 is 6431159051
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 35
Digital Root 8

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