How to convert $\arctan(24/7) $to $2\arctan(3/4)$?

Question

This is actually a doubt I got while solving this question. The thing is I know how to convert $2\arctan(3/4)$ to $\arctan(24/7)$ by using the $\arctan x + \arctan y$ identity, but how do I do the opposite? Please help!

Answer 1

$$2\arctan\frac34\\=\arctan\left(\tan\left(2\arctan\frac34\right)\right) \\=\arctan\left(\frac{2\tan\left(\arctan\dfrac34\right)}{1-\tan^2\left(\arctan\dfrac34\right)}\right) \\=\arctan\left(\frac{\dfrac32}{1-\dfrac9{16}}\right) \\=\arctan\frac{24}7$$

can be read top-down or bottom-up !

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To discover the bottom-up formula, you need to solve

$$\frac{2y}{1-y^2}=x$$

which is a quadratic equation.

$$xy^2+2y-x=0$$ has the solutions

$$y=\frac{1\pm\sqrt{1+x^2}}{x}.$$

For the solutions to be rational, you need to use Pythagorean triples such as $(24,7)$ and

$$y=\frac{1+\dfrac{\sqrt{24^2+7^2}}7}{\dfrac{24}7}=\frac 43.$$

Answer 2

Hint: $\displaystyle\tan(2x)=\frac{2\tan x}{1-\tan^2x}$.

Answer 3

Note that $(4+3i)^2 = 7+24i$ and take args.