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!
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1Please type your questions using MathJax. – N. F. Taussig Oct 05 '18 at 10:11
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Hint: $\displaystyle\tan(2x)=\frac{2\tan x}{1-\tan^2x}$.
José Carlos Santos
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Would you please give a bit more detail? Though I can do it the reverse way I have no idea how to do the forward proof, and what to take as what based on this formula. Also according to the question I did there is no way of knowing beforehand what the RHS is to be. – Hema Oct 05 '18 at 11:24
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1@Hema Use this equality to compute $\tan\left(2\arctan\left(\frac34\right)\right)$. – José Carlos Santos Oct 05 '18 at 11:27
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$$2\arctan\frac34\\=\arctan\left(\tan\left(2\arctan\frac34\right)\right) \\=\arctan\left(\frac{2\tan\left(\arctan\dfrac34\right)}{1-\tan^2\left(\arctan\dfrac34\right)}\right) \\=\arctan\left(\frac{\dfrac32}{1-\dfrac9{16}}\right) \\=\arctan\frac{24}7$$
can be read top-down or bottom-up !
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
$$y=\frac{1+\dfrac{\sqrt{24^2+7^2}}7}{\dfrac{24}7}=\frac 43.$$
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1Bottom up you need intuition. You need to know what arctan is. You need to imagine that you need some sort of Pythagoras theorem and look for it. Of course, you don't have to go bottom up: if you start by $2\arctan y=\arctan \frac{24}{7}$ you can look for $y$ in a more standard way. – orion Oct 05 '18 at 11:24
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