If sec a=cos a + sin a

Question

how do we prove that cos 2a= [tan(pi/2 - a) ]^2 ?

Answer 1

if
1/cos a = cos a + sin a
1 = oos^2 a + sin a oos a
cos^2 a = 1 - sin a cos a Needed for last step

if
sin (pi/2 -a ) = cos a
cos (pi/2- a) = sin a
so
tan(pi/2 - a) = cos a/sin a
so
[tan(pi/2 - a) ]^2 = cos^2 a/sin^2 a

cos 2a = cos^2 a - sin^2 a identity
so
prove
cos^2a/sin^2a = 1/sin^2 a - cos a/sin a
cos^2a = 1 - sin a cos a ---true

Answer 2

Thanks a lot!!