Nested formulas for cosine and inverse cosine functions based on Vi\`ete's formula for $\pi$

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.