Continued fraction expansion of tan(x)

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Introduction

One of the most intriguing formulas which I encountered during my 12th grade was the continued fraction expansion of $ \tan x$. It was given in a book on "Numerical Analysis" and was offered as an example of a formula which converges very fast (like something comparable to the Maclaurin series for $ \sin x$ and $ \cos x$). However like the usual practice followed in mathematics textbooks it was offered without any proof or any context. I checked the formula using calculator and was amazed with its speed of convergence. Just 10 terms were enough to give the result with an accuracy up to 10 decimals.