Use reference angles to find the exact value of the expression cos17π/4. Show all your work.
To find the exact value of the expression $\cos\left(\frac{17\pi}{4}\right)$, we can use reference angles and the unit circle.
First, we determine the reference angle. Since $\frac{17\pi}{4}$ is in quadrant IV, we can determine the reference angle by subtracting the closest multiple of $\frac{\pi}{2}$ (which is $\frac{8\pi}{4} = 2\pi$) from $\frac{17\pi}{4}$.
$\text{Reference angle} = \frac{17\pi}{4} - 2\pi = \frac{\pi}{4}$
Next, we recall that $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.
Since $\cos$ is even, we can determine the value of $\cos(\frac{17\pi}{4})$ by reflecting the value of $\frac{\pi}{4}$ about the $y$-axis. This gives us:
$\cos(\frac{17\pi}{4}) = \cos(-\frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \boxed{\frac{\sqrt{2}}{2}}$.