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	one billion eight hundred fifty-seven million five hundred twelve thousand eighty-nine

Interesting facts

1857512089 has 16 divisors, whose sum is 2040923520
The reverse of 1857512089 is 9802157581
Previous prime number is 331

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 46
Digital Root 1

Name

Name	one billion three hundred ninety-five million two hundred seventy-one thousand fifty

Interesting facts

1395271050 has 32 divisors, whose sum is 4018381056
The reverse of 1395271050 is 0501725931
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 33
Digital Root 6

Name

Name	two hundred twenty-five million two hundred forty-four thousand one hundred seventy-one

Interesting facts

225244171 has 8 divisors, whose sum is 233532800
The reverse of 225244171 is 171442522
Previous prime number is 229

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 28
Digital Root 1

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