Verify the Identity tan(theta)^2cos(theta)^2=1/(csc(theta)^2)

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Trigonometry

$\tan^{2} (θ) \cos^{2} (θ) = \frac{1}{\csc^{2} (θ)}$

Start on the left side.

$\tan^{2} (θ) \cos^{2} (θ)$

Convert to sines and cosines.

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

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

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

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

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

$\sin^{2} (θ)$

Now consider the right side of the equation.

$\frac{1}{\csc^{2} (θ)}$

Convert to sines and cosines.

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

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

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

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

$1 \frac{{\sin (θ)}^{2}}{1^{2}}$

Multiply $\frac{{\sin (θ)}^{2}}{1^{2}}$ by $1$ .

$\frac{{\sin (θ)}^{2}}{1^{2}}$

One to any power is one.

$\frac{{\sin (θ)}^{2}}{1}$

Divide ${\sin (θ)}^{2}$ by $1$ .

$\sin^{2} (θ)$

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

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

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Name

Name	one billion eight hundred sixty-three million four hundred ten thousand four hundred fifty-eight

Interesting facts

1863410458 has 64 divisors, whose sum is 3613059072
The reverse of 1863410458 is 8540143681
Previous prime number is 163

Basic properties

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

Name

Name	one billion six hundred sixty-one million three hundred fourteen thousand two hundred fifty-two

Interesting facts

1661314252 has 16 divisors, whose sum is 4025492352
The reverse of 1661314252 is 2524131661
Previous prime number is 13

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 31
Digital Root 4

Name

Name	eight hundred million six hundred seventy-four thousand six hundred fifty-one

Interesting facts

800674651 has 16 divisors, whose sum is 1055380480
The reverse of 800674651 is 156476008
Previous prime number is 109

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 37
Digital Root 1

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