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.
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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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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