Question

How do you prove `cos2t=(1-tan^2 t)/(1+tan^2 t)`?

Answer 1

Remembering that `sec^2(t)-1=tan^2(t)` and the identity
`cos(2t) = cos^2(t) - sin^2(t)` as well as each of the reciprocal identities, namely

`tan t = sin t / cos t` and `sec t = 1/cos t`, we can start off by simplifying our equation on the right-hand side.

`cos(2t) = (1-sin^2(t)/(cos^2(t)))/(1/cos^2(t)) = (1/1 * cos^2(t)/1)/(cancel(1/cos^2(t) * cos^2(t)/1)) - (sin^2(t)/(cancel(cos^2(t))) * cancel(cos^2(t)/1))/(1/cancel(cos^2(t)) * cancel(cos^2(t)/1))`

`= cos^2(t) - sin^2(t)`