Find the set of values, which are taken by tan⁡z I was trying to write tangents as follows: tan⁡(z)=−ieiz−e−izeiz+e−iz and then z=a+bi , which led me to tan⁡z=−icos⁡a(e−b−eb)+isin⁡a(e−b+eb)cos⁡a(e−b+eb)+isin⁡a(e−b−eb)

Question

Find the set of values, which are taken by tan⁡z  I was trying to write tangents as follows:  tan⁡(z)=−ieiz−e−izeiz+e−iz  and then   z=a+bi   , which led me to  tan⁡z=−icos⁡a(e−b−eb)+isin⁡a(e−b+eb)cos⁡a(e−b+eb)+isin⁡a(e−b−eb)

izumrledk · Accepted Answer

It is much easier to deal with this problem using the fact that  1+tan2(z)=1cos2(z)  So, which numbers can be written as 1cos2(z)? Answer: all, except 0. It follows from this (and from the fact that tan⁡ is an odd function), that the range of tan⁡  is  C{±i}

Jude Carpenter · Accepted Answer

HINT  We have that by the standard trick  tan⁡(z)=−ieiz−e−izeiz+e−iz⋅eiz―+e−iz―eiz―+e−iz―=…  then use the well know identities for cos,sin,cosh  and  sinh