Prove that tan^2x/(1+tan^2x) = sin^2x

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marizi | High School Teacher | (Level 1) Associate Educator

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Prove the trigonometric identity `(tan^2(x))/(1+tan^2(x)) = sin^2(x)`

Substitute the trigonometric identity `tan^2(x) = sec^2(x)-1`

Note : This is the same as `1 +tan^2(x) = sec^2(x).`

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

Express it into two fractions:

`=(sec^2(x))/(sec^2(x)) - 1/(sec^2(x))`

Simplify and substitute   ` cos^2(x)=1/(sec^2(x))` :

`= 1 - cos^2(x)`

`= sin^2(x) `                

Note `1-cos^2(x) = sin^2(x)` is the same as `sin^2(x) +cos^2(x) =1`

This proves `(tan^2(x))/(1+tan^2(x)) = sin^2(x)`

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Prove the identity ` ``(tan^2x)/(1+tan^2x)=sin^2x ` :

Note that `tan^2x+1=sec^2x=1/(cos^2x) ` and `tan^2x=(sin^2x)/(cos^2x) `

Substituting we get:

`(tan^2x)/(1+tan^2x)=((sin^2x)/(cos^2x))/(1/(cos^2x))=sin^2x ` as required.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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

Start with the left hand side.

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

Substitute `tanx = sin x/cos x`

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

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

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

Use the fact `sin^2x + cos^2x = 1`

= `sin^2x/1`

= `sin^2x `

This proves `(tan^2x)/(1+tan^2x) = sin^2x`

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