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Prove this trig identity: `(1-tan^2(x/2))/(1+tan^2(x/2))=cosx`

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tempaccount2 | Honors

Posted April 25, 2013 at 2:23 AM via web

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Prove this trig identity: `(1-tan^2(x/2))/(1+tan^2(x/2))=cosx`

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oldnick | Valedictorian

Posted April 25, 2013 at 2:51 AM (Answer #1)

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`(1-tan^2(x/2))/(1+tan^2(x/2))=(1-(sin^2(x/2))/(cos^2(x/2)))/(1+(sin^2(x/2))/(cos^2(x/2)))` `=((cos^2(x/2)-sin^2(x/2))/(cos^2(x/2)))/((cos^2(x/2)+sin^2(x/2))/(cos^2(x/2)))` `=(cos^2sin(x/2)-sin^2(x/2))/(cos^2(x/2)+cos^2(x/2))` `=cos(2(x/2))/1=cosx`

From formuila:

 

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oldnick | Valedictorian

Posted April 25, 2013 at 2:56 AM (Answer #2)

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ERRATA CORRIGE AT THE SECOND ROW YOU'D READ:

`(cos^2(x/2)-sin^2(x/2))/(sin^2(x/2)+cos^2(x/2))=cos(2x/2)/1=cosx`

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted April 25, 2013 at 3:54 AM (Answer #4)

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The trigonometric identity `(1-tan^2(x/2))/(1+tan^2(x/2))=cosx` has to be proved.

`(1-tan^2(x/2))/(1+tan^2(x/2))`

= `(1-(sin^2(x/2))/(cos^2(x/2)))/(1+(sin^2(x/2))/(cos^2(x/2)))`

Use the identity `sin^2x + cos^2x = 1` and `cos^2x - sin^2x = cos(2x)`

= `((cos x)/(cos^2(x/2)))/(1/(cos^2(x/2)))`

= cos x

This proves that `(1-tan^2(x/2))/(1+tan^2(x/2))=cosx`

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