Answer
$cos(x-y)=\frac{1}{9}(4\sqrt {2}+\sqrt 5)$
Work Step by Step
$sec~x=\frac{3}{2}$
$cos~x=\frac{1}{sec~x}=\frac{2}{3}$
$sin^2x=1-cos^2x=1-(\frac{2}{3})^2=1-\frac{4}{9}=\frac{5}{9}$
$sin~x=\frac{\sqrt 5}{3}~~$ (Quadrant I)
$csc~y=3$
$sin~y=\frac{1}{csc~y}=\frac{1}{3}$
$cos^2y=1-sin^2y=1-(\frac{1}{3})^2=1-\frac{1}{9}=\frac{8}{9}$
$cos~y=\frac{2\sqrt 2}{3}~~$ (Quadrant I)
Use the addition formula for sine.
$cos(x-y)=cos~x~cos~y+sin~x~sin~y$
$cos(x-y)=\frac{2}{3}~\frac{2\sqrt 2}{3}+\frac{\sqrt 5}{3}\frac{1}{3}$
$cos(x-y)=\frac{4\sqrt {2}+\sqrt 5}{9}=\frac{1}{9}(4\sqrt {2}+\sqrt 5)$
