Can anyone verify this identity: cos2t=(1-tan^2 t/1+tan^2 t)?

Question

I need to verify this identity.

cos 2t = (1-tan^2 t / 1+tan^2 t)

What do you guys think? Can anyone help, PLEASE!

Answer 1

Yes I can It was a pain in the a** but here is how you go about it
cos(2t)=cos(t)^2-sin(t)^2 (double angle identity)

now you have to divide the right hand side by cos(t)^2 and multiply by cos(t)^2 or in other words multiply by one.
making
cos(2t)=(1-tan(t)^2)(cos(t)^2)

now all you have to prove is that the cos(t)^2 also represents 1/(1+tan(t)^2)

which comes from the other trig identity that sec(t)^2=1+tan(t)^2 ... this identity comes from cosx^2+sinx^2=1

Answer 2

A = 1 - sin²t / cos²t = (cos²t - sin²t) / cos²t
B = 1 + sin²t / cos²t = (cos²t + sin²t) /cos²t
B = 1 / cos²t

A / B = (cos²t - sin²t) = cos 2t as required.