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

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Trigonometry

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

Start on the left side.

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

Apply Pythagorean identity in reverse.

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

Convert to sines and cosines.

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

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

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

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

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

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

One to any power is one.

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

Simplify the denominator.

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Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Add $1$ and $1$ .

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

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

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

Combine the numerators over the common denominator.

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

Combine.

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

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

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

Combine $\cos (x)$ and $\frac{- 1 + 2 {\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

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

Reduce the expression $\frac{\cos (x) (- 1 + 2 {\cos (x)}^{2})}{{\cos (x)}^{2}}$ by cancelling the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Combine $\sin (x)$ and $\frac{\cos (x)}{- 1 + 2 {\cos (x)}^{2}}$ .

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Factor $- 1$ out of $2 {\cos (x)}^{2}$ .

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

Factor $- 1$ out of $- 1 (1) - (- 2 {\cos (x)}^{2})$ .

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

Move the negative in front of the fraction.

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

Write $- 1$ as a fraction with denominator $1$ .

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

Combine.

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

Remove parentheses.

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

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

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

Move the negative in front of the fraction.

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

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

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

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

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

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Name

Name	four hundred thirty million seven hundred ninety thousand four hundred

Interesting facts

430790400 has 4096 divisors, whose sum is 15554713824
The reverse of 430790400 is 004097034
Previous prime number is 81

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 27
Digital Root 9

Name

Name	one billion six hundred seventy million nine hundred sixty-one thousand eight hundred fifty-six

Interesting facts

1670961856 has 256 divisors, whose sum is 19047312000
The reverse of 1670961856 is 6581690761
Previous prime number is 1471

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 49
Digital Root 4

Name

Name	one billion five hundred seventy-one million five hundred fifty thousand seven hundred eighty-one

Interesting facts

1571550781 has 8 divisors, whose sum is 1663407200
The reverse of 1571550781 is 1870551751
Previous prime number is 181

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 40
Digital Root 4

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