if tanh (x/2) = tan(x/2) then show that cosh x cosx = 1.

cos (x) = (1- tan ^2 (x/2))/(1+ tan ^2 (x/2))
= (1- tanh ^2(x/2))/( 1+ tanh^2 (x/2)) as tan (x/2) = tanh (x2)
= ( cosh^(x/2) - sinh^2 (x/2))/( cosh^(x/2) + sinh^2 (x/2))
= 1/ cosh x

so cosh x cosx = 1.

proved