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

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Trigonometry

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

Start on the left side.

$\sin (x) \tan (x) + \cos (x) - \sec (x) + 1$

Simplify the expression.

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Simplify each term.

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Rewrite $\tan (x)$ in terms of sines and cosines.

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Combine the numerators over the common denominator.

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

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Rewrite $- 1$ as $- 1 (1)$ .

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

Factor $- 1$ out of $\sin^{2} (x)$ .

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

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

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

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Apply pythagorean identity.

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

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

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

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

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 $1$ .

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

Combine the opposite terms in $\cos (x) + 1 - \cos (x)$ .

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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.

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

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Name

Name	four hundred seventy-seven million six hundred two thousand four hundred two

Interesting facts

477602402 has 4 divisors, whose sum is 716403606
The reverse of 477602402 is 204206774
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 32
Digital Root 5

Name

Name	seven hundred thirty-one million one hundred forty-nine thousand two hundred forty-nine

Interesting facts

731149249 has 4 divisors, whose sum is 746705664
The reverse of 731149249 is 942941137
Previous prime number is 47

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 40
Digital Root 4

Name

Name	one hundred eighty-five million two hundred fifty-three thousand sixty-five

Interesting facts

185253065 has 4 divisors, whose sum is 222303684
The reverse of 185253065 is 560352581
Previous prime number is 5

Basic properties

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

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