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

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