Prove that $4\tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right)= \frac{\pi}{4}$
Prove that $4\tan^{-1} \left(\dfrac{1}{5}\right) - \tan^{-1}\left(\dfrac{1}{239}\right)=\dfrac{\pi}{4}.$
I was wondering if there was a shorter solution than the method below?
Below is my attempt using what I would call the standard approach to these kinds of problems.
The expression on the left hand side is equivalent to $$\tan^{-1}\left[\tan \left(4\tan^{-1}\left(\dfrac{1}{5}\right)\right)-\tan^{-1}\left(\dfrac{1}{239}\right)\right]\\ =\tan^{-1}\left(\dfrac{\tan(4\tan^{-1}(\frac{1}{5}))-\frac{1}{239}}{1+\frac{1}{239}\tan(4\tan^{-1}(\frac{1}{5}))}\right)\;(1).$$
We have that $\tan(4\tan^{-1}(\frac{1}{5}))=\dfrac{2\tan(2\tan^{-1}(\frac{1}{5}))}{1-\tan^2(2\tan^{-1}(\frac{1}{5})}\;(2)$
and that
$\tan(2\tan^{-1}(\frac{1}{5}))=\dfrac{2\cdot \frac{1}{5}}{1-(\frac{1}{5})^2}=\dfrac{5}{12}\;(3).$
Plugging in the result of $(3)$ into $(2)$ gives $$\tan\left(4\tan^{-1}\left(\frac{1}{5}\right)\right) = \dfrac{2\cdot \frac{5}{12}}{1-(\frac{5}{12})^2}=\dfrac{120}{119}\;(4).$$
Pluggin in the result of $(4)$ into $(1)$ gives that the original expression is equivalent to$\tan^{-1}\left(\dfrac{\frac{120}{119}-\frac{1}{239}}{1+\frac{1}{239}\cdot\frac{120}{119}}\right)=\tan^{-1}\left(\dfrac{\frac{119\cdot 239 + 239-119}{239\cdot 119}}{\frac{119\cdot 239+120}{119\cdot 239}}\right)=\tan^{-1}(1)=\dfrac\pi4,$
as desired.
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$\begingroup$Shortest proof:
$$(5+i)^4(239-i)=114244+114244i.$$
Taking the arguments,
$$4\arctan \frac15-\arctan\frac1{239}=\frac\pi4.$$
Note that the computation avoids the fractions and immediately generalizes to other Machin-like formulas ().
To perform the computation by hand, consider
$$(5+i)^2=24+10i\propto12+5i,$$
$$(12+5i)^2=119+120i,$$
$$(119+120 i)(239-i)=(119\cdot239+120)+(120\cdot239-119)i\propto 1+i.$$
(After simplification by $119\cdot239$, we have $120=239-119$.)
$\endgroup$ $\begingroup$We can also use
$$\arctan(u) \pm \arctan(v) = \arctan\left(\frac{u \pm v}{1 \mp uv}\right)$$
to obtain in four steps
$$\frac{\frac15 - \frac1{239}}{1 + \frac1{5\cdot 239}}=\frac{239-5}{5\cdot 239+1}=\frac{234}{5\cdot 239+1}=\frac9{46} \to$$
$$\to \frac{\frac15 + \frac9{46}}{1 - \frac15\frac9{46}}= \frac7{17} \\\to \frac{\frac15 + \frac7{17}}{1 - \frac15\frac7{17}}= \frac2{3} \\\to \frac{\frac15 + \frac2{3}}{1 - \frac15\frac2{3}}= 1$$
$\endgroup$ 2 $\begingroup$A slightly faster variant of the same computation using the identity $$\tan^{-1} u \pm \tan^{-1} v = \tan^{-1} \frac{u \pm v}{1 \mp u v}$$ can be performed by observing that in the special case $u = v$ $$2\tan^{-1} u = \tan^{-1} \frac{2u}{1-u^2}.$$ Consequently, we iterate $g(u) = 2u/(1-u^2)$ twice for $u = 1/5$ to obtain $$4 \tan^{-1} \frac{1}{5} = \tan^{-1} g(g(\tfrac{1}{5})) = \tan^{-1} \frac{120}{119}.$$ Now we apply the original formula to obtain $$4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239} = \tan^{-1} \frac{\frac{120}{119} - \frac{1}{239}}{1 + \frac{120}{(119)(239)}} = \tan^{-1} 1 = \frac{\pi}{4}.$$ In all, we used three steps instead of four.
It is also worth noting that when $u, v \in \mathbb Q$, we can write $$\tan^{-1} \frac{p}{q} \pm \tan^{-1} \frac{r}{s} = \tan^{-1} \frac{ps \pm qr}{qs \mp pr}.$$ If we think of each rational as being represented by an ordered pair, which in turn is an element of the complex numbers, e.g. $u = p/q$ has the representation $z = q + pi$, and we define the function $$T(z,w) = \tan\left(\tan^{-1} \frac{\Im(z)}{\Re(z)} + \tan^{-1} \frac{\Im(w)}{\Re(w)}\right),$$ then $$T(z,w) = \frac{\Im(zw)}{\Re(zw)}.$$ In fact, the inverse tangent identity is simply a consequence of multiplication in the complex plane: $$\arg(zw) = \arg(z) + \arg(w).$$ I leave the details of this relationship as an exercise for the reader.
From the above, we may then regard Machin's formula as a statement about the existence of a nonzero real number $\rho$ such that $$(5+i)^4 = \rho(1+i)(239+i).$$ What is this number?
$\endgroup$ $\begingroup$As advised by Maximilian Janisch, you should use the $\tan x$ formula rather $\tan^{-1}x$:$$\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right) - \tan^{-1}\left(\dfrac{1}{239}\right)\right]=\tan\left[\dfrac{\pi}{4}\right] \iff \\ \frac{\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right)\right]-\frac1{239}}{1+\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right)\right]\cdot \frac1{239}}=1 \iff \\ \tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right)\right]=\frac{120}{119} \iff \\ \frac{2\tan\left[2\tan^{-1} \left(\dfrac{1}{5}\right)\right]}{1-\tan^2\left[2\tan^{-1} \left(\dfrac{1}{5}\right)\right]}=\frac{120}{119} \iff \\ \frac{2\cdot \frac{2\cdot \frac15}{1-\frac1{5^2}}}{1-\left[\frac{2\cdot \frac15}{1-\frac1{5^2}}\right]^2}=\frac{120}{119} \iff \\ \frac{\frac5{6}}{1-\frac{25}{144}}=\frac{120}{119} \ \checkmark$$
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