What is tan (cos^-1 (2/3)) equal to?

Since the arccosine function returns the vaue of the
angle, we'll write arccos (2/3) = t => cos t =
2/3

We'll write the tangent function as a
fraction:

tan t = sin t/ cos
t

tan t = sin (arccos (2/3))/cos (arccos
(2/3))

We know that cos (arccos x) = x and sin (arccos x) =
sqrt (1 - x^2)

tan t = sqrt(1 -
4/9)/(2/3)

tan t =
sqrt5/3/(2/3)

We'll simplify and we'll
get:

tan t = (sqrt
5)/2

The required value of tan (cos ^-1(2/3))
is (sqrt 5)/2.