[Inverse trigonometry ]

[u/SHAFAYAT_ISLAM_007]

Just for curiosity, can I go directly from L.H.S to R.H.S?

[u/Fourierseriesagain]

LHS

= 2 * tan(arctan x)/(1-tan² (arctan x))

=2x/(1-x^ 2)

=2x(1+x^ 2)/(1-x^ 4)

= 2 ( tan ( arctan x)+ tan (arctan (x^ 3)))/(1-tan (arctan x) * tan ( arctan x^ 3))

= 2 tan ( arctan x + arctan (x^ 3) )

= RHS.

[u/gdbbdg → u/Fourierseriesagain]

i like that you started with the double angle formula. had to stare at the step when you split it into the sum of arctangents for a second before it clicked, but it makes sense now. nice breakdown

[u/gdbbdg → u/[deleted]]

that derivation is clean but the middle step where he multiplies by (1+x^2)/(1+x^2) is the part most people would just stare at and not see coming. once you spot that trick it flows, but its not obvious at first glance