a) Find all solutions for $\tan (x)=0$. b) If $\sin (x)=\frac{1}{3}$ and $\sec (y)=\frac{5}{4}$, where $0 \leq x \leq \frac{\pi}{2}$ and $0 \leq y \leq \frac{\pi}{2}$ , evaluate the expression $\cos (x-y)$.

Step 1 ：The tangent function is zero at multiples of \(\pi\). So, the solutions to \(\tan (x)=0\) are \(x=n\pi\), where \(n\) is an integer.

Step 2 ：Let's find the solutions for the range of \(n\) from -10 to 10. The solutions are approximately [-31.42, -28.27, -25.13, -21.99, -18.85, -15.71, -12.57, -9.42, -6.28, -3.14, 0.0, 3.14, 6.28, 9.42, 12.57, 15.71, 18.85, 21.99, 25.13, 28.27, 31.42].

Step 3 ：To evaluate \(\cos (x-y)\), we need to find the values of \(x\) and \(y\). We know that \(\sin (x)=\frac{1}{3}\) and \(\sec (y)=\frac{5}{4}\). We can find \(x\) and \(y\) by taking the inverse sine and cosine respectively.

Step 4 ：By calculating, we get \(x\) approximately equals to 0.34 and \(y\) approximately equals to 0.64.

Step 5 ：Then, we can substitute these values into the expression for \(\cos (x-y)\).

Step 6 ：By calculating, we get \(\cos (x-y)\) approximately equals to 0.954.

Step 7 ：\(\boxed{\text{Final Answer:}}\)

Step 8 ：a) The solutions for \(\tan (x)=0\) are \(x=n\pi\), where \(n\) is an integer. For the range of \(n\) from -10 to 10, the solutions are approximately [-31.42, -28.27, -25.13, -21.99, -18.85, -15.71, -12.57, -9.42, -6.28, -3.14, 0.0, 3.14, 6.28, 9.42, 12.57, 15.71, 18.85, 21.99, 25.13, 28.27, 31.42].

Step 9 ：b) The value of \(\cos (x-y)\), where \(\sin (x)=\frac{1}{3}\) and \(\sec (y)=\frac{5}{4}\), is approximately 0.954.