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How to prove that modulus of  `[x(1-x^2)]/(1+x^2)^2` is less 0.25 with help...

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lenalen | Student, Undergraduate | eNoter

Posted November 23, 2011 at 6:08 PM via web

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How to prove that modulus of  `[x(1-x^2)]/(1+x^2)^2` is less 0.25 with help of  trigonometry?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted November 24, 2011 at 4:58 AM (Answer #1)

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`|(x(1-x^2))/(1+x^2)^2|lt0.25`

Use the formula:

sin 2t =`( 2tan t)/(1+tan^2 t)`

cos 2t = `(1- tan^2 t)/(1 + tan^2 t)`

 Substitute`(1-x^2)/(1+x^2)^2`  by cos`2 alpha`

Multiply `x/(1+x^2) by 2 =gt |2x/(1+x^2)|lt 2*0.25`

Substitute `2x/(1+x^2)`  by sin`2 alpha` .

Write the modulus using sin`2 alpha`  and cos `2 alpha` .

`|sin 2 alpha* cos 2 alpha|lt 2*0.25`

Multiply by 2:

`|2sin 2 alpha* cos 2 alpha|lt 2*2*0.25`

Identify the formula of sine of double angle:

`|sin 2*(2 alpha)|lt 1`

ANSWER: The last inequality proves the identity

`|x(1-x^2)/(1+x^2)^2|lt0.25`

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