A nice proof using complex numbers, and also the fact that e^{iπ} = -1. (where we have taken the principal value)pic.twitter.com/MFdFktIOgH
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There is more arctan fun to be had. Solve the trig equation I posted earlier using two more methods and derive all sorts of identities:pic.twitter.com/nTSFhheh21
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That’s just a trick.
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A little proof; arctan(1) = π/4, then arctan(2) + arctan(3) = 3π/4 and tan(x) is a function means tan(arctan(2) + arctan(3)) = tan(3π/4) = -1. Finally, tan(arctan(2) + arctan(3)) = (tan(arctan(2)) + tan(arctan(3)))/(1 - tan(arctan(2))tan(arctan(3))) = 5/-5 = -1 QED
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