Question

How do you simplify `sec(theta/2)-tan(theta/2)` using the double angle identities?

Answer 1

`=1/cos(theta/2)+sin(theta/2)/cos(theta/2)=(1+sin(theta/2))/cos(theta/2)=(sin(theta/4)+cos(theta/4))^2/(cos^2(theta/4)-sin^2(theta/4))`
`=(cos(theta/4)+sin(theta/4))/(cos(theta/4)-sin(theta/4))`
`=(1+tan(theta/4))/(1-tan(theta/4))`
`=(tan(pi/4)+tan(theta/4))/(1-tan(pi/4)xxtan(theta/4))`
`=tan(pi/4+theta/4)`