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

justaguide | College Teacher | (Level 2) Distinguished Educator

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

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

Use the fact that `tan x = sin x/cos x`

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

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

`cos^2x + sin^2x = 1`

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

=>` sin^2x`

As shown above `(tan^2 x) / (1 + tan^2 x) = sin^2x` not tan x.

The given relation is not an identity. It holds for only a few angles not as a general rule.

lochana2500 | Student, Undergraduate | (Level 1) Valedictorian

Posted on

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tan²x ÷ (1+tan²x) ≠ tanx

tan²x ÷ (1+tan²x) = sin²x

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L:H:S ≡ tan²x ÷ (1+tan²x)

= tan²x ÷ sec²x

= (sin²x/cos²x) ÷ 1/cos²x

= sin²x

= R:H:S
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