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)