Prove that $$\frac{\sin 40^\circ}{\sin 80^\circ} +\frac{\sin 80^\circ}{\sin 20^\circ} -\frac{\sin 20^\circ}{\sin 40^\circ} =3.$$
I tried taking the least common multiple of the denominator and then simplifying the numerator but I do not get the value $3$.
Best Answer
Just for fun, a complicated proof by dissection:
The sine theorem gives that the length of the red segment is $\frac{\sin 40}{\sin 80}$, hence:
$$ 3+\frac{\sin 20}{\sin 40} = \frac{\sin 80}{\sin 20}+\frac{\sin 40}{\sin 80}$$
as wanted.