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

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Trigonometry

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

Start on the left side.

$\cot (x) (\cot (x) + \tan (x))$

Apply the distributive property.

$\cot (x) \cot (x) + \cot (x) \tan (x)$

Multiply $\cot (x) \cot (x)$ .

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

Apply Pythagorean identity in reverse.

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

Convert to sines and cosines.

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

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

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

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

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

Simplify.

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Simplify each term.

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One to any power is one.

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

Cancel the common factor of $\cos (x)$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $\sin (x)$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Add $- 1$ and $1$ .

$\frac{1}{{\sin (x)}^{2}} + 0$

Add $\frac{1}{{\sin (x)}^{2}}$ and $0$ .

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

Now consider the right side of the equation.

$\csc^{2} (x)$

Convert to sines and cosines.

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

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

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

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

One to any power is one.

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

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

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

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Name

Name	four hundred ninety-five million seven hundred ninety-eight thousand four hundred fifty-seven

Interesting facts

495798457 has 16 divisors, whose sum is 619517952
The reverse of 495798457 is 754897594
Previous prime number is 463

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 58
Digital Root 4

Name

Name	one billion eight hundred eighty-six million four hundred ninety-three thousand nine hundred twelve

Interesting facts

1886493912 has 64 divisors, whose sum is 8523071424
The reverse of 1886493912 is 2193946881
Previous prime number is 251

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 51
Digital Root 6

Name

Name	one billion five hundred eighteen million eight hundred nineteen thousand five hundred twenty-eight

Interesting facts

1518819528 has 32 divisors, whose sum is 6834687984
The reverse of 1518819528 is 8259188151
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 48
Digital Root 3

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Convert to Polar Coordinates (-5,3)