Let $x$ be an angle such that $\tan x = \frac{a}{b}$ and $\tan 2x = \frac{b}{a + b}.$  Then the least positive value of $x$ equals $\tan^{-1} k.$  Compute $k.$
Question:
Let $x$ be an angle such that $\tan x = \frac{a}{b}$ and $\tan 2x = \frac{b}{a + b}.$  Then the least positive value of $x$ equals $\tan^{-1} k.$  Compute $k.$
Answer:
Let's think step by step.

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Let $x$ be an angle such that $\tan x = \frac{a}{b}$ and $\tan 2x = \frac{b}{a + b}.$  Then the least positive value of $x$ equals $\tan^{-1} k.$  Compute $k.$
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