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

Start on the left side.

To write <math><mstyle displaystyle="true"><mfrac><mrow><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow><mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math> .

To write <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn></mrow><mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn></mrow></mfrac></mstyle></math> .

Write each expression with a common denominator of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> , by multiplying each by an appropriate factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Combine.

Combine.

Reorder the factors of <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> .

Combine the numerators over the common denominator.

Simplify each term.

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Apply the distributive property.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⋅</mo><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> from <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Add <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> and <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>sec</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math> using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Apply pythagorean identity.

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

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Name | eight hundred forty-six million one hundred sixty-eight thousand eighty-eight |
---|

- 846168088 has 64 divisors, whose sum is
**2951931600** - The reverse of 846168088 is
**880861648** - Previous prime number is
**59**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 49
- Digital Root 4

Name | eight hundred fifty-five million seven hundred eighty-eight thousand four hundred seventy |
---|

- 855788470 has 16 divisors, whose sum is
**1540775808** - The reverse of 855788470 is
**074887558** - Previous prime number is
**6367**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 52
- Digital Root 7

Name | eight hundred seventy-nine million four hundred thousand six hundred seventeen |
---|

- 879400617 has 8 divisors, whose sum is
**896372136** - The reverse of 879400617 is
**716004978** - Previous prime number is
**593**

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