Answer
\begin{align*}
&\sin\theta = \frac{15}{17} \enspace \cos\theta = \frac{8}{17} \enspace \tan\theta = \frac{15}{8} \\
&\csc\theta = \frac{17}{15} \enspace \sec\theta = \frac{17}{8} \enspace \cot\theta = \frac{8}{15}
\end{align*}
Work Step by Step
Let $x$ be the length of unknown side of right angled triangle. According to Pythagorean theorem, $15^2+x^2 = 17^2$
\[\implies x^2 = 17^2-15^2\]
\[x^2 = 64\]
\[x = 8\]
\begin{align*}
&\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \enspace \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \enspace \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \\
&\csc\theta = \frac{\text{hypotenuse}}{\text{opposite}} \enspace \sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}} \enspace \cot\theta = \frac{\text{adjacent}}{\text{opposite}}
\end{align*}
\begin{align*}
\implies &\sin\theta = \frac{15}{17} \enspace \cos\theta = \frac{8}{17} \enspace \tan\theta = \frac{15}{8} \\
&\csc\theta = \frac{17}{15} \enspace \sec\theta = \frac{17}{8} \enspace \cot\theta = \frac{8}{15}
\end{align*}
