Simplify 7\arctan^2\phi+2\arctan^2\phi^3-\arctan^2\phi^5

Gage Potter

Gage Potter

Answered question

2022-04-30

Simplify
7arctan2ϕ+2arctan2ϕ3arctan2ϕ5

Answer & Explanation

eslasadanv3

eslasadanv3

Beginner2022-05-01Added 20 answers

A well-known formula for the sum of arctangents will be used here:
arctanu+arctanv=arctan(u+v1uv) (modπ)
The exact equality holds for uv<1, for other values there is additional term — an integer multiple of π
We can determine the following identities using this formula:
arctanϕ=π212arctan2
arctanϕ3=π4+12arctan2
arctanϕ5=π32arctan2
Expanding parenthesis and plugging these into the question's original form allows us to see that all arctan2 terms, and we get the result
7arctan2ϕ+2arctan2ϕ3arctan2ϕ5=7π28

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