Simple trig shows that
tan(OAD) = tan(40°)*tan(30°)*tan(20°)
and any scietific calculator will show that angle OAD = 10° to
the accuracy of the calculator.
However, the standard formula for
tan(3*x) = (3*tan(x) - (tan(x)3))/(1-3*(tan(x))3)
Let t = tan(x) and T = tan(3*x) then
t3-3*T*t2 - 3*t + T = 0
If x = 10° then this is a cubic equation whose roots are tan(10°)
, tan(130°) and tan(250°).
Minus the product of these will be the constant
term T = tan(30°).
That is -tan(30°) = tan(10°) * tan(130°)* tan(250°)
But tan(130°) = -tan(50°) and tan(250°) = tan(70°)
from which the result follows.
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