This question is from my assignment in complex analysis and I was unable to solve it.
Prove that tanz doesn't assume the value i, -i. Does this contradict Picard's Theorem.
Attempt- tanz = $ i \frac{e^{iz} -e^{-iz}} { e^{iz}+ e^{-iz}}$ => $e^{iz}=i$ and z=π/2 and z=-π/2 is not in domain. But how does it tells which values will never be attained? So, how to deduce it?
I think picard theorem is not violated as tanz is not entire, due to cosz being in denominator.
Am I right?
No, you want to say that $$\tan z=i\frac{e^{iz}-e^{-iz}}{e^{iz}+ e^{-iz}}=\pm i \,,$$ which becomes equivalent to $$e^{\pm iz}=0\,,$$ which is impossible (the exponential function has no complex zeroes).