1

tan^{-1}\left(  \frac{1}{2} \right) 

tan^{-1}\left(  \frac{1}{3} \right) 

\frac{π}{4}

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Inverse Trigonometric Functions
A Visualisation for

tan^{-1}\left( \frac{1}{2}  \right) +tan^{-1}\left( \frac{1}{3}  \right)=\frac{π}{4}

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Angle between a side and diagonal in a Square is 45°

by
Benny Varghese
PGT Maths
JNV Pathanamthitta

Another One

tan^{-1}(1)

tan^{-1}(1)+tan^{-1}\left( \frac{1}{2}  \right) +tan^{-1}\left( \frac{1}{3}  \right)=\frac{π}{2}

A = (1, 0)
B = (3, 0)
f = Segment(A, B)
C = (3, 1)
h = Segment(C, A)
D = (5, 2)
i = Segment(C, D)
E = (7, 3)
j = Segment(D, E)
F = (6, 5)
k = Segment(E, F)
l = Segment(F, A)
a: 0
A' = Dilate(A, If[0 < a ≤ 1, 1 - a, 0], B)
m = Segment(A, A')
B' = Dilate(B, If[1 < a ≤ 2, 2 - a, 0], C)
g = Segment(B, B')
c = Circle(A, 1)
H = Intersect(c, m)
I = Intersect(c, h)
AH = Distance(A, H)
n = Segment(A, H)
α = Angle(H, A, I)
A'_1 = Rotate(A, If[2 < a < 3, ((3 - a) * α), α], A)
H' = Rotate(H, If[2 < a < 3, ((a - 2) * α), α], A)
n' = Segment(A'_1, H')
d = CircleSector(A, H, H')
A'_2 = Dilate(A, If[3 < a ≤ 4, 4 - a, 0], E)
p = Segment(A, A'_2)
E' = Dilate(E, If[4 < a ≤ 5, 5 - a, 0], F)
q = Segment(E, E')
G = Intersect(c, l)
β = Angle(I, A, G)
r = Segment(A, H')
H'_1 = Rotate(H, If[2 < a < 3, ((a - 2) * α), α], A)
J = (1, 2)
K = Point(l)
q1, j_1, k_1, b, a_1 = Polygon(J, K, B, A)
q2, j_2, k_2, b_1, a_2 = Polygon(J, K, B, A)
γ = Angle(C, B, A)
δ = Angle(E', E, A)
A'_3 = Dilate(A, If[5 < a < 6, 6 - a, 0], F)
s = Segment(A, A'_3)
t = Segment(A, I)
e = Segment(A, I)
A'_4 = Rotate(A, If[6 < a < 7, ((a - 6) * β), β], A)
I' = Rotate(I, If[6 < a < 7, ((a - 6) * β), β], A)
e' = Segment(A'_4, I')
c_1 = CircleSector(A, I, I')
len = If(3.66 > a > 3.33, 1, If[3.99 > a > 3.66, 2, 3])
text1: "" + len + ""
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stroke2 = PolyLine((2.1125214150512126, 0.2566032388394786), (NaN, NaN), (2.1125214150512126, 0.2566032388394786), (2.172759477489543, 0.2566032388394786), (2.2209499274402074, 0.2566032388394786), (2.2570927649032058, 0.23250801386414643), (2.281187989878538, 0.19636517640114814), (2.30528321485387, 0.16022233893814986), (2.329378439829202, 0.11203188898748546), (2.3534736648045342, 0.07588905152448717), (2.3655212772922005, 0.03974621406148888), (2.3775688897798664, 0.003603376598490585), (2.4016641147551985, -0.03253946086450771), (2.4137117272428648, -0.0807299108151721), (2.425759339730531, -0.11687274827817039), (2.437806952218197, -0.1530155857411687), (2.461902177193529, -0.18915842320416698), (2.4739497896811953, -0.22530126066716527), (2.4980450146565274, -0.28553932310549573), (2.5100926271441932, -0.3216821605684941), (2.534187852119526, -0.36987261051915843), (2.558283077094858, -0.4060154479821567), (2.606473527045522, -0.442158285445155), (2.618521139533188, -0.4783011229081533), (2.6546639769961864, -0.5144439603711516), (2.6787592019715185, -0.5505867978341499), (2.714902039434517, -0.5867296352971482), (2.7510448768975153, -0.5987772477848143), (2.7871877143605133, -0.6349200852478126), (2.8233305518235117, -0.6590153102231447), (2.85947338928651, -0.683110535198477), (2.895616226749508, -0.695158147686143), (2.9317590642125064, -0.7072057601738092), (2.967901901675505, -0.7313009851491413), (3.016092351626169, -0.7313009851491413), (3.0642828015768333, -0.7313009851491413), (3.1124732515274975, -0.7313009851491413), (NaN, NaN), (2.991997126650837, -0.6349200852478126), (NaN, NaN), (2.991997126650837, -0.6349200852478126), (3.016092351626169, -0.6710629227108109), (3.040187576601501, -0.7072057601738092), (3.076330414064499, -0.7313009851491413), (3.1124732515274975, -0.7433485976368075), (3.148616088990496, -0.7674438226121396), (3.1004256390398317, -0.7674438226121396), (3.0642828015768333, -0.7794914350998058), (3.028139964113835, -0.803586660075138), (2.991997126650837, -0.8276818850504701), (2.9558542891878385, -0.8517771100258024), (2.9317590642125064, -0.8879199474888007), true)
o: 0
opacity: o
f_1 = Segment(C, K)
L = Point(f_1)
g_1 = Segment(L, J)
M = Point(g_1)
h_1 = Segment(M, A)
N = (7.485756584550287, 2.4613163240823743)
O = (11.750611405184086, 2.4010782616440443)
P = Intersect(c, h_1)
d_1 = CircleSector(A, G, P)
u = false
stroke3 = PolyLine((1.4137598907665798, 0.9794599880994445), (NaN, NaN), (1.4137598907665798, 0.9794599880994445), (1.4017122782789138, 1.0276504380501088), (1.3896646657912477, 1.0637932755131072), (1.3776170533035816, 1.0999361129761054), (1.3655694408159154, 1.1481265629267698), (1.3414742158405832, 1.1842694003897682), (1.293283765889919, 1.2204122378527664), (1.2571409284269206, 1.2445074628280985), (1.2089504784762564, 1.2445074628280985), (1.1607600285255921, 1.2445074628280985), (1.1125695785749277, 1.2445074628280985), (1.0764267411119295, 1.2324598503404325), (1.0402839036489313, 1.2083646253651004), (1.004141066185933, 1.1963170128774343), (0.9679982287229347, 1.160174175414436), (0.9318553912599364, 1.1481265629267698), (0.9077601662846043, 1.1119837254637717), (0.871617328821606, 1.0758408880007733), (0.8475221038462739, 1.039698050537775), (0.8113792663832756, 1.0035552130747767), (0.7993316538956096, 0.9553647631241123), (0.7872840414079434, 0.919221925661114), (0.7752364289202773, 0.8830790881981158), (0.7511412039449452, 0.8469362507351175), (0.7390935914572792, 0.8107934132721192), (0.727045978969613, 0.7626029633214547), (0.7029507539942809, 0.7264601258584564), (0.6909031415066148, 0.6903172883954581), (0.6788555290189486, 0.6541744509324598), (0.6668079165312826, 0.6180316134694616), (0.6547603040436165, 0.5818887760064633), (0.6186174665806182, 0.545745938543465), (0.58247462911762, 0.5096031010804667), (0.5463317916546218, 0.4855078761051345), (0.5101889541916235, 0.4614126511298023), (0.4619985042409592, 0.4373174261544701), (0.42585566677796094, 0.4132222011791379), (0.38971282931496265, 0.4011745886914718), (0.34152237936429836, 0.4011745886914718), (0.30537954190130007, 0.3770793637161396), (NaN, NaN), (0.3776652168272966, 0.5698411635187972), (NaN, NaN), (0.3776652168272966, 0.5698411635187972), (0.3776652168272966, 0.5216507135681328), (0.36561760433963053, 0.4855078761051345), (0.34152237936429836, 0.44936503864213617), (0.32947476687663224, 0.4132222011791379), (0.30537954190130007, 0.3770793637161396), (0.26923670443830183, 0.3409365262531413), (0.24514147946296966, 0.304793688790143), (0.20899864199997142, 0.28069846381481084), (0.24514147946296966, 0.2686508513271447), (0.293331929413634, 0.2686508513271447), (0.32947476687663224, 0.28069846381481084), (0.3776652168272966, 0.28069846381481084), (0.42585566677796094, 0.28069846381481084), (0.47404611672862523, 0.28069846381481084), (0.5222365666792896, 0.28069846381481084), (0.58247462911762, 0.28069846381481084), true)