Question

If a is any real number the number of roots of cot x-tanx=ain the first quadrant is

Answer 1

cot(x) - tan(x) = a ----> (1/tan(x)) - tan(x) = a ----> 1 - tan^2(x) = a*tan(x) ---->

tan^2(x) + a*tan(x) - 1 = 0. Using the quadratic formula we have

tan(x) = (-a +/- sqrt(a^2 + 4))/2 = (-a/2) +/- sqrt((a/2)^2 + 1).

To have a root in the first quadrant we require that tan(x) > 0.

If a = 0 then we have the one root x = pi/4.

If a > 0 then only tan(x) = (-a/2) + sqrt((a/2)^2 + 1) gives us a desired root.

If a < 0 then only tan(x) = (-a/2) + sqrt((a/2)^2 + 1) gives us a desired root.

Thus we will always have 1 root in the first quadrant