  In this article, we develop nested representations for cosine and inverse
cosine functions, which is a generalization of Vi\`{e}te's formula for $\pi$.
We explore a natural inverse relationship between these representations and
develop numerical algorithms to compute them. Throughout this article, we
perform numerical computation for various test cases, and demonstrate that
these nested formulas are valid for complex arguments and a $k$th branch. We
further extend the presented results to hyperbolic cosine and logarithm
functions, and using additional trigonometric identities, we explore the sine
and tangent functions and their inverses.
