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