Simple trig shows that
cot(ABY) = cot(30°) + cot(40°) - cot(80°)
A scientific calculator will show that angle ABY is 20° to the accuracy of the calculator.
Using the fact that tan(90 - x) = cot(x) shows that we must prove
tan(70°) = tan(60°) + tan(50°) - tan(10°)
However, using the standard formula
tan(3*x) = (3*tan(x) - (tan(x)3))/(1-3*(tan(x))2).
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°).
The sum of these roots will be minus the coefficient of t2 which is 3 * tan(30°) = tan(60°).
tan(60°) = tan(10°)- tan(50°) + tan(70°).
From which the result follows.
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