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

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

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vaaruni | Student

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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