Verify the Identity (csc(-x))/(sec(-x))=-cot(x)

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Trigonometry

$\frac{\csc (- x)}{\sec (- x)} = - \cot (x)$

Start on the left side.

$\frac{\csc (- x)}{\sec (- x)}$

Since $\csc (- x)$ is an odd function, rewrite $\csc (- x)$ as $- \csc (x)$ .

$\frac{- \csc (x)}{\sec (- x)}$

Since $\sec (- x)$ is an even function, rewrite $\sec (- x)$ as $\sec (x)$ .

$\frac{- \csc (x)}{\sec (x)}$

Convert to sines and cosines.

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

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

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

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

Combine.

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

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

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

Move the negative in front of the fraction.

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

Now consider the right side of the equation.

$- \cot (x)$

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

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

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

$\frac{\csc (- x)}{\sec (- x)} = - \cot (x)$ is an identity

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Name

Name	four hundred fourteen million three hundred ninety-seven thousand sixty-eight

Interesting facts

414397068 has 32 divisors, whose sum is 950223744
The reverse of 414397068 is 860793414
Previous prime number is 997

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 42
Digital Root 6

Name

Name	one billion ninety-seven million one hundred twenty-eight thousand three hundred seventy-eight

Interesting facts

1097128378 has 16 divisors, whose sum is 1699729920
The reverse of 1097128378 is 8738217901
Previous prime number is 2341

Basic properties

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

Name

Name	three hundred forty-three million seven hundred seventy-five thousand five hundred ninety-six

Interesting facts

343775596 has 16 divisors, whose sum is 788089824
The reverse of 343775596 is 695577343
Previous prime number is 53

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 49
Digital Root 4

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