Answer
$cos2y=\frac{7}{25}$
Work Step by Step
Evaluate the expression $cos2y$
Given: $sinx=\frac{1}{3}$ and $secy=\frac{5}{4}$
Since, $cos2y=cos^{2}y-sin^{2}y$ ...(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
$cos2y=cos^{2}y-sin^{2}y$
$=(\frac{4}{5})^{2}-(\frac{3}{5})^{2}$
$=\frac{16}{25}-\frac{9}{25}$
Hence, $cos2y=\frac{7}{25}$
