Verify the Identity tan(x)+cot(x)=sec(x)csc(x)

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Trigonometry

$\tan (x) + \cot (x) = \sec (x) \csc (x)$

Start on the left side.

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

Convert to sines and cosines.

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

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

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

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

Add fractions.

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

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

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

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

Write each expression with a common denominator of $\cos (x) \sin (x)$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Combine.

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

Reorder the factors of $\sin (x) \cos (x)$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

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

Apply pythagorean identity.

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

Now consider the right side of the equation.

$\sec (x) \csc (x)$

Convert to sines and cosines.

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

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

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

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

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

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

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

$\tan (x) + \cot (x) = \sec (x) \csc (x)$ is an identity

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Name

Name	one billion one hundred twenty-one million nine hundred eighteen thousand one hundred

Interesting facts

1121918100 has 128 divisors, whose sum is 3598912512
The reverse of 1121918100 is 0018191211
Previous prime number is 607

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 24
Digital Root 6

Name

Name	eight hundred fifty-eight million three hundred forty-one thousand nine hundred thirty-one

Interesting facts

858341931 has 8 divisors, whose sum is 1144607184
The reverse of 858341931 is 139143858
Previous prime number is 10457

Basic properties

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

Name

Name	one billion one hundred forty-nine million nine hundred twelve thousand four hundred fifty-nine

Interesting facts

1149912459 has 8 divisors, whose sum is 2044288832
The reverse of 1149912459 is 9542199411
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 45
Digital Root 9

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