SOLUTION: Without using the calculator find 1/(sqrt(3)-tan (20°))+1/(sqrt(3)+tan(40°))+1/(sqrt(3)-tan(80°))

This problem is very difficult.  I will only get it in terms of sin(20°).
Then it will involve the solving the cubic equation for sin(3q), which can be done, although it's a pain in the butt.





Simplifying by multiplying each fraction by the LCD of each denominator:



Change all the cos(60°)'s to 1/2 and

Simplify by multiplying each numerator and denominator by 2, and sin(-20°) to -sin(20°)





Get all the denominators to 4sin(20°) so we can combine the numerators 
over the common denominator:













20° is not a special angle; however, 60° is, and 60° = 3∙20°.

So look up the formula for sin(3q) and solve for sin(q) 


Edwin