Let <math><mstyle displaystyle="true"><mi>u</mi><mo>=</mo><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> . Substitute <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> for all occurrences of <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Factor using the perfect square rule.

Rewrite <math><mstyle displaystyle="true"><mn>4</mn><msup><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mn>2</mn><mi>u</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Check the middle term by multiplying <math><mstyle displaystyle="true"><mn>2</mn><mi>a</mi><mi>b</mi></mstyle></math> and compare this result with the middle term in the original expression.

Simplify.

Factor using the perfect square trinomial rule <math><mstyle displaystyle="true"><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> , where <math><mstyle displaystyle="true"><mi>a</mi><mo>=</mo><mn>2</mn><mi>u</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi><mo>=</mo><mn>1</mn></mstyle></math> .

Replace all occurrences of <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> with <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> .

Replace the left side with the factored expression.

Set the factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from both sides of the equation.

Divide each term by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and simplify.

Divide each term in <math><mstyle displaystyle="true"><mn>2</mn><mi>csc</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Cancel the common factor.

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

Move the negative in front of the fraction.

The range of cosecant is <math><mstyle displaystyle="true"><mi>y</mi><mo>≤</mo><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mi>y</mi><mo>≥</mo><mn>1</mn></mstyle></math> . Since <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> does not fall in this range, there is no solution.

No solution

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Name | two billion eighty-one million two hundred five thousand eleven |
---|

- 2081205011 has 4 divisors, whose sum is
**2081897400** - The reverse of 2081205011 is
**1105021802** - Previous prime number is
**3019**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 20
- Digital Root 2

Name | one billion seven hundred sixteen million seven hundred sixty-three thousand fifty |
---|

- 1716763050 has 16 divisors, whose sum is
**2591256000** - The reverse of 1716763050 is
**0503676171** - Previous prime number is
**199**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 36
- Digital Root 9

Name | sixty-nine million two hundred sixty thousand four hundred ninety-three |
---|

- 69260493 has 8 divisors, whose sum is
**92914528** - The reverse of 69260493 is
**39406296** - Previous prime number is
**163**

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