Start on the left side.

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

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

Write each expression with a common denominator of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><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><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><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>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Simplify each term.

Apply pythagorean identity.

Simplify the numerator.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mo>-</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mrow><mo>(</mo><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> out of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

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

Factor <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> out of <math><mstyle displaystyle="true"><mo>-</mo><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> out of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⋅</mo><mn>1</mn><mo>+</mo><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>1</mn><mo>-</mo><mi>cos</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Now consider the right side of the equation.

Apply the reciprocal identity to <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mi>sin</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac></mstyle></math> .

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

Do you know how to Verify the Identity (1-cos(x))/(sin(x))+(sin(x))/(1-cos(x))=2csc(x)? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

Name | one billion four hundred sixty-seven million four hundred forty-two thousand five hundred seventy-three |
---|

- 1467442573 has 16 divisors, whose sum is
**1554973056** - The reverse of 1467442573 is
**3752447641** - Previous prime number is
**83**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 43
- Digital Root 7

Name | one billion three hundred fifty-two million three hundred fifty-eight thousand eight hundred seventy-two |
---|

- 1352358872 has 32 divisors, whose sum is
**4626736632** - The reverse of 1352358872 is
**2788532531** - Previous prime number is
**73**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 44
- Digital Root 8

Name | nine hundred forty-eight million four hundred forty-three thousand six hundred ninety-four |
---|

- 948443694 has 8 divisors, whose sum is
**1465776720** - The reverse of 948443694 is
**496344849** - Previous prime number is
**33**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 51
- Digital Root 6