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

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Trigonometry

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

Start on the left side.

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

Apply Pythagorean identity in reverse.

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

Simplify.

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Combine the numerators over the common denominator.

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

Subtract $1$ from $1$ .

$\cos (x) + 1 + \frac{- {\cos (x)}^{2} + 0}{\cos (x)}$

Add $- {\cos (x)}^{2}$ and $0$ .

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

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

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Factor $\cos (x)$ out of $- {\cos (x)}^{2}$ .

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

Cancel the common factors.

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Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- \cos (x)$ by $1$ .

$\cos (x) + 1 - \cos (x)$

Subtract $\cos (x)$ from $\cos (x)$ .

$0 + 1$

Add $0$ and $1$ .

$1$

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

$\sec^{2} (x) \cos^{2} (x)$

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

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

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Name

Name	one hundred forty-six million four hundred seventy thousand seven hundred thirty-one

Interesting facts

146470731 has 8 divisors, whose sum is 213048384
The reverse of 146470731 is 137074641
Previous prime number is 11

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 33
Digital Root 6

Name

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

Interesting facts

1793892775 has 8 divisors, whose sum is 2583205632
The reverse of 1793892775 is 5772983971
Previous prime number is 5

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 58
Digital Root 4

Name

Name	one billion four hundred eighty-nine million one hundred fifty-nine thousand five hundred sixty-four

Interesting facts

1489159564 has 32 divisors, whose sum is 3499739136
The reverse of 1489159564 is 4659519841
Previous prime number is 1087

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 52
Digital Root 7

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