If sec theta + tan theta=m; find sin theta
Hmm. I was looking for a cute way, but I guess I'll just have to brute-force it.
secx + tanx = m
secx(1+sinx) = m
1+sinx = m cosx
(1+sinx)^2 = m^2 cos^2 x
sin^2 + 2sin + 1 = m^2 - m^2 sin^2
(1+m^2)sin^2 + 2sin + 1 = 0
now solve that using the quadratic formula.
Because of the squaring, check for extraneous roots.