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 -...

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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))` **