Verify the Identity (cos(x)sin(x))/(cot(x))=1-cos(x)^2

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Trigonometry

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

Start on the left side.

$\frac{\cos (x) \sin (x)}{\cot (x)}$

Simplify the expression.

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Separate fractions.

$\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\cot (x)}$

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{1} \cdot \frac{\cos (x)}{\frac{\cos (x)}{\sin (x)}}$

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\sin (x)}{1} (\cos (x) \frac{\sin (x)}{\cos (x)})$

Write $\cos (x)$ as a fraction with denominator $1$ .

$\frac{\sin (x)}{1} (\frac{\cos (x)}{1} \cdot \frac{\sin (x)}{\cos (x)})$

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

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Cancel the common factor.

$\frac{\sin (x)}{1} (\frac{\cos (x)}{1} \cdot \frac{\sin (x)}{\cos (x)})$

Rewrite the expression.

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

Divide $\sin (x)$ by $1$ .

$\sin (x) \sin (x)$

Multiply $\sin (x) \sin (x)$ .

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Raise $\sin (x)$ to the power of $1$ .

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

Raise $\sin (x)$ to the power of $1$ .

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

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

${\sin (x)}^{1 + 1}$

Add $1$ and $1$ .

$\sin^{2} (x)$

Apply Pythagorean identity in reverse.

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

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

$\frac{\cos (x) \sin (x)}{\cot (x)} = 1 - \cos^{2} (x)$ is an identity

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Name

Name	one hundred thirty-seven million five hundred fifty-six thousand nine hundred seventy-eight

Interesting facts

137556978 has 16 divisors, whose sum is 296276736
The reverse of 137556978 is 879655731
Previous prime number is 13

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 51
Digital Root 6

Name

Name	one billion five hundred sixty-one million one hundred fifteen thousand one hundred ten

Interesting facts

1561115110 has 16 divisors, whose sum is 2932181856
The reverse of 1561115110 is 0115111651
Previous prime number is 23

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 22
Digital Root 4

Name

Name	one billion eight hundred sixty-four million seven hundred thirty-eight thousand six hundred forty

Interesting facts

1864738640 has 128 divisors, whose sum is 11497399200
The reverse of 1864738640 is 0468374681
Previous prime number is 67

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 47
Digital Root 2

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