Answer
$\frac{\sqrt 5}{5}$
Work Step by Step
Let $cos^{-1}\frac{2}{3}=y$, we have $cos(y)=\frac{2}{3}, $ and
$tan(\frac{1}{2}cos^{-1}\frac{2}{3})=tan(\frac{y}{2})
=\sqrt {\frac{1-cos(y)}{1+cos(y)}}=\sqrt {\frac{1-2/3}{1+2/3}}=\frac{\sqrt 5}{5}$
Precalculus: Mathematics for Calculus, 7th Edition
by Stewart, James; Redlin, Lothar; Watson, Saleem
Published by
Brooks Cole
ISBN 10:
1305071751
ISBN 13:
978-1-30507-175-9
Chapter 7 - Section 7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas - 7.3 Exercises - Page 561: 50
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