Question

How do you verify the identity `sec(u/2)=+-sqrt((2tanu)/(tanu+sinu))`?

Answer 1

Square both sides of the equation
`(2tan u)/(tan u + sin u) = sec^2 (u/2)`
Develop the left side:
`LS = ((2sinu)/(cos u))/(sin u/(cos u) + sin u) =`

`= (2sin u)/(sin u + sin u.cos u) = (2sin u)/((sin u)(1 + cos u))`

Replace `( 1 + cos u)` by `2cos^2 (u/2)`, we get:

`LS = 1/cos^2 (u/2) = sec^2 (u/2)`