The equality $ \cos\eta+i\sin\eta = \frac{1+\cos\theta+c\cdot i\sin\theta}{1+\cos\theta-c\cdot i\sin\theta} \tag{1} $ or equivalently $ \frac{1+\cos\eta+\frac1c\cdot i\sin\eta}{1+\cos\eta-\frac1c\cdot i\sin\eta} = \cos\theta+i\sin\theta \tag{1} $ holds precisely if $\tan\frac\eta2 = c\cdot\tan\frac\theta2.\tag{2}$
Does this appear in any published tabulation of trigonometric identities, or is it otherwise "known"?
PS: My original posting mangled $(1)$ so that it said something that didn't make sense and was simpler than what should have been there.