Verify the Identity (csc(theta)^2)/(csc(theta)^2-1)=sec(theta)^2

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Trigonometry

$\frac{\csc^{2} (θ)}{\csc^{2} (θ) - 1} = \sec^{2} (θ)$

Start on the left side.

$\frac{\csc^{2} (θ)}{\csc^{2} (θ) - 1}$

Apply pythagorean identity.

$\frac{\csc^{2} (θ)}{\cot^{2} (θ)}$

Convert to sines and cosines.

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

$\frac{{(\frac{1}{\sin (θ)})}^{2}}{\cot^{2} (θ)}$

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

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

Apply the product rule to $\frac{1}{\sin (θ)}$ .

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

Apply the product rule to $\frac{\cos (θ)}{\sin (θ)}$ .

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

Combine.

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

Cancel the common factor of ${\sin (θ)}^{2}$ .

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Cancel the common factor.

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

Rewrite the expression.

$\frac{1^{2}}{{\cos (θ)}^{2}}$

One to any power is one.

$\frac{1}{\cos^{2} (θ)}$

Rewrite $\frac{1}{\cos^{2} (θ)}$ as $\sec^{2} (θ)$ .

$\sec^{2} (θ)$

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

$\frac{\csc^{2} (θ)}{\csc^{2} (θ) - 1} = \sec^{2} (θ)$ is an identity

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Name

Name	three hundred ninety-six million eight hundred nineteen thousand nine

Interesting facts

396819009 has 8 divisors, whose sum is 464116000
The reverse of 396819009 is 900918693
Previous prime number is 19

Basic properties

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

Name

Name	one hundred seven million three hundred nine thousand four hundred fifty

Interesting facts

107309450 has 32 divisors, whose sum is 232104960
The reverse of 107309450 is 054903701
Previous prime number is 1471

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 29
Digital Root 2

Name

Name	nine hundred ten million nine hundred twenty-three thousand eight hundred thirty

Interesting facts

910923830 has 8 divisors, whose sum is 1639662912
The reverse of 910923830 is 038329019
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 35
Digital Root 8

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