Verify the Identity (1-sin(t))/(cos(t)^2)+1/(1-sin(t))=2sec(t)^2

$\frac{1 - \sin (t)}{\cos^{2} (t)} + \frac{1}{1 - \sin (t)} = 2 \sec^{2} (t)$

Start on the left side.

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

Simplify the expression.

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To write $\frac{1 - \sin (t)}{\cos^{2} (t)}$ as a fraction with a common denominator, multiply by $\frac{1 - \sin (t)}{1 - \sin (t)}$ .

$\frac{1 - \sin (t)}{\cos^{2} (t)} \cdot \frac{1 - \sin (t)}{1 - \sin (t)} + \frac{1}{1 - \sin (t)}$

To write $\frac{1}{1 - \sin (t)}$ as a fraction with a common denominator, multiply by $\frac{\cos^{2} (t)}{\cos^{2} (t)}$ .

$\frac{1 - \sin (t)}{\cos^{2} (t)} \cdot \frac{1 - \sin (t)}{1 - \sin (t)} + \frac{1}{1 - \sin (t)} \cdot \frac{\cos^{2} (t)}{\cos^{2} (t)}$

Write each expression with a common denominator of $(1 - \sin (t)) \cos^{2} (t)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{1 - \sin (t)}{\cos^{2} (t)}$ and $\frac{1 - \sin (t)}{1 - \sin (t)}$ .

$\frac{(1 - \sin (t)) (1 - \sin (t))}{\cos^{2} (t) (1 - \sin (t))} + \frac{1}{1 - \sin (t)} \cdot \frac{\cos^{2} (t)}{\cos^{2} (t)}$

Multiply $\frac{1}{1 - \sin (t)}$ and $\frac{\cos^{2} (t)}{\cos^{2} (t)}$ .

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

Reorder the factors of $(1 - \sin (t)) \cos^{2} (t)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Expand $(1 - \sin (t)) (1 - \sin (t))$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- \sin (t)) - \sin (t) (1 - \sin (t)) + \cos^{2} (t)}{\cos^{2} (t) (1 - \sin (t))}$

Apply the distributive property.

$\frac{1 \cdot 1 + 1 (- \sin (t)) - \sin (t) \cdot 1 - \sin (t) (- \sin (t)) + \cos^{2} (t)}{\cos^{2} (t) (1 - \sin (t))}$

Simplify and combine like terms.

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Simplify each term.

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Multiply $1$ by $1$ .

$\frac{1 + 1 (- \sin (t)) - \sin (t) \cdot 1 - \sin (t) (- \sin (t)) + \cos^{2} (t)}{\cos^{2} (t) (1 - \sin (t))}$

Multiply $- \sin (t)$ by $1$ .

$\frac{1 - \sin (t) - \sin (t) \cdot 1 - \sin (t) (- \sin (t)) + \cos^{2} (t)}{\cos^{2} (t) (1 - \sin (t))}$

Multiply $- 1$ by $1$ .

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

Multiply $- \sin (t) (- \sin (t))$ .

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Multiply $- 1$ by $- 1$ .

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

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

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

Raise $\sin (t)$ to the power of $1$ .

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

Raise $\sin (t)$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Subtract $\sin (t)$ from $- \sin (t)$ .

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

Rewrite $1 - 2 \sin (t) + \sin^{2} (t) + \cos^{2} (t)$ in a factored form.

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Apply pythagorean identity.

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

Add $1$ and $1$ .

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

Factor $2$ out of $2 - 2 \sin (t)$ .

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Factor $2$ out of $2$ .

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

Factor $2$ out of $- 2 \sin (t)$ .

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

Factor $2$ out of $2 (1) + 2 (- \sin (t))$ .

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

Cancel the common factor of $1 - \sin (t)$ .

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Cancel the common factor.

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

Rewrite the expression.

$\frac{2}{\cos^{2} (t)}$

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

$2 \sec^{2} (t)$

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

$\frac{1 - \sin (t)}{\cos^{2} (t)} + \frac{1}{1 - \sin (t)} = 2 \sec^{2} (t)$ is an identity

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Name

Name	sixty-four million nine hundred sixty thousand seven hundred fifty-one

Interesting facts

64960751 has 4 divisors, whose sum is 65111904
The reverse of 64960751 is 15706946
Previous prime number is 431

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 38
Digital Root 2

Name

Name	one billion six hundred two million four hundred seventy-eight thousand one

Interesting facts

1602478001 has 4 divisors, whose sum is 1603215600
The reverse of 1602478001 is 1008742061
Previous prime number is 2179

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 29
Digital Root 2

Name

Name	two hundred sixty-five million ninety-three thousand nine hundred fifty-three

Interesting facts

265093953 has 8 divisors, whose sum is 353997600
The reverse of 265093953 is 359390562
Previous prime number is 659

Basic properties

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

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