Answer
$cos(x-y)=\frac{1}{15}(8\sqrt 2+3)$
Work Step by Step
Evaluate the expression $cos(x-y)$
Given: $sinx=\frac{1}{3}$ and $secy=\frac{5}{4}$
$cos(x-y)=cosxcosy+sinxsiny$ ...(1)
Thus,
$sinx=\frac{1}{3}$ gives opp =1, hyp = 3 and adj $=\sqrt {3^{2}-1^{2}}=2\sqrt 2$
Therefore, $cos x=\frac{2\sqrt 2}{3}$
Now, $secy=\frac{5}{4}$ gives hyp =5, adj =4
and opp $=\sqrt {5^{2}-3^{2}}=3$
Therefore, $siny=\frac{3}{5}$ and $cosy=\frac{4}{5}$
Equation (1) becomes
$cos(x-y)=(\frac{2\sqrt 2}{3})(\frac{4}{5})+(\frac{1}{3})(\frac{3}{5})$
$=\frac{8\sqrt 2}{15}+\frac{3}{15}$
Hence, $cos(x-y)=\frac{1}{15}(8\sqrt 2+3)$