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Show that `tan^2 x = (1 - cos(2x))/(1 + cos(2x))`

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xetatheta | Student, Kindergarten | Salutatorian

Posted January 22, 2013 at 8:32 AM via web

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Show that `tan^2 x = (1 - cos(2x))/(1 + cos(2x))`

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted January 22, 2013 at 8:37 AM (Answer #1)

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

Use the relation `cos(2x) = 2*cos^2x - 1 = 1 - 2*sin^2x`

 `(1 - cos(2x))/(1 + cos(2x))`

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

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

=> `(2*sin^2x)/(2*cos^2x)`

=> `(sin^2x)/(cos^2x)`

=> `((sin x)/(cos x))^2`

=> `tan^2x`

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

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vaaruni | High School Teacher | Salutatorian

Posted January 22, 2013 at 2:34 PM (Answer #2)

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We are require to prove :-  tan^2(x)= (1-cos2x)/(1+cos2x)

Let us take R.H.S -> (1-cos2x)/(1+cos2x)

(1-cos2x)/(1+cos2x)= {1-(1-2sin^2(2x))}/{(1+(2cos^2(2x)-1)}

[Using formula:- cos(2A)=1-2sin^2(A)   Or  cos(2A)=2cos^2(A)-1 ]

=> (1+cos2x)/(1-cos2x)=(1-1+2sin^2(x))/((1+2cos^2(x)-1))

=> (1+cos2x)/(1-cos2x)= 2sin^2(x)/2cos^2(x)

=> (1+cos2x)/(1-cos2x)= tan^2(x)   <-- Proved

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