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

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Trigonometry

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

Start on the left side.

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

Apply Pythagorean identity.

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Apply pythagorean identity.

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

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

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

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

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

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

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

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

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

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

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

Apply pythagorean identity.

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

Convert to sines and cosines.

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

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

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

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

Simplify.

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One to any power is one.

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

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

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

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

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

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

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

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Name

Name	one billion one hundred fourteen million forty-seven thousand five hundred nineteen

Interesting facts

1114047519 has 8 divisors, whose sum is 1209826080
The reverse of 1114047519 is 9157404111
Previous prime number is 17

Basic properties

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

Name

Name	one billion one hundred seventy-seven million eight hundred fifteen thousand seven hundred eighty-nine

Interesting facts

1177815789 has 16 divisors, whose sum is 1745568000
The reverse of 1177815789 is 9875187711
Previous prime number is 3343

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 54
Digital Root 9

Name

Name	one hundred fifty-seven million forty-five thousand two hundred ninety-six

Interesting facts

157045296 has 128 divisors, whose sum is 1072854720
The reverse of 157045296 is 692540751
Previous prime number is 83

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 39
Digital Root 3

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