Question

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

Answer 1

`cos(theta/2)-2tan(theta/2)=sqrt((1+costheta)/2)-2sqrt((1-costheta)/(1+costheta))`

Explanation:

As `cos2A=2cos^2A-1`, `cos(theta/2)=sqrt((1+costheta)/2)` - here consider `A=theta/2`

also `cos2A=1-2sin^2A`, hence similarly we have

`sin(theta/2)=sqrt((1-costheta)/2)`

Hence `cos(theta/2)-2tan(theta/2)` `cos(theta/2)-2tan(theta/2)`

= `sqrt((1+costheta)/2)-2(sqrt((1-costheta)/2))/(sqrt((1+costheta)/2))`

= `sqrt((1+costheta)/2)-2sqrt((1-costheta)/(1+costheta))`