Multiple Angle Identities for Tangent

2. (Double angle identity for tangent)

Multiple Angle Identities for Hyperbolic Tangent

2. (Double angle identity for hyperbolic tangent)

For real x ≠ 0 and each positive n,

• .

It is obvious that we acknowledge the existence of complex number in mathematics.  However, we still have no clues about authentic natural essences of complex number from our current mathematic systems. Complex number is an ironic wall to restrict what human can see authentic faces of mathematics from its nature.

What is "Some Virtual Thoughts" ?  Some Virtual Thoughts is a section to list the comments that reflect on any events or forms. This section is not a math resource.  There are no right or wrong words in terms of events or forms found in this section because no judgments or arguments are considered. Why is this section posted in here?  We post this topic because it satisfies the curiousness about the observations of some persons.