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

t^{3}-3*T*t^{2} - 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 t^{2}
which is 3 * tan(30°) = tan(60°).

Thus

tan(60°) = tan(10°)- tan(50°) + tan(70°).

From which the result follows.

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