Verify sec(x)-tan(x)sin(x)=1/(sec(x))

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Trigonometry

$\sec (x) - \tan (x) \sin (x) = \frac{1}{\sec (x)}$

Simplify $\sec (x) - \tan (x) \sin (x)$ .

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Simplify terms.

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Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Combine the numerators over the common denominator.

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

Apply pythagorean identity.

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

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

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

$\frac{\cos (x) \cos (x)}{\cos (x)} = \frac{1}{\sec (x)}$

Cancel the common factors.

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Multiply by $1$ .

$\frac{\cos (x) \cos (x)}{\cos (x) \cdot 1} = \frac{1}{\sec (x)}$

Cancel the common factor.

$\frac{\cos (x) \cos (x)}{\cos (x) \cdot 1} = \frac{1}{\sec (x)}$

Rewrite the expression.

$\frac{\cos (x)}{1} = \frac{1}{\sec (x)}$

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

$\cos (x) = \frac{1}{\sec (x)}$

Simplify $\frac{1}{\sec (x)}$ .

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

$\cos (x) = \frac{1}{\frac{1}{\cos (x)}}$

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$\cos (x) = 1 \cos (x)$

Multiply $\cos (x)$ by $1$ .

$\cos (x) = \cos (x)$

For the two functions to be equal, the arguments of each must be equal.

$x = x$

Move all terms containing $x$ to the left side of the equation.

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Subtract $x$ from both sides of the equation.

$x - x = 0$

Subtract $x$ from $x$ .

$0 = 0$

Since $0 = 0$ , the equation will always be true.

Always true

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Name

Name	forty-two million one hundred ninety-six thousand five hundred eighty-four

Interesting facts

42196584 has 64 divisors, whose sum is 201056256
The reverse of 42196584 is 48569124
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 8
Sum of Digits 39
Digital Root 3

Name

Name	sixty-five million nine hundred forty-eight thousand four hundred twenty-one

Interesting facts

65948421 has 32 divisors, whose sum is 111200256
The reverse of 65948421 is 12484956
Previous prime number is 71

Basic properties

Is Prime? no
Number parity odd
Number length 8
Sum of Digits 39
Digital Root 3

Name

Name	one billion seven hundred sixteen million three hundred nine thousand one hundred fifteen

Interesting facts

1716309115 has 16 divisors, whose sum is 2097538560
The reverse of 1716309115 is 5119036171
Previous prime number is 7747

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 34
Digital Root 7

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