Multiple Angle Identities for Cosine

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.