Answer
The value of the trigonometric function $\cos \left( -\frac{\pi }{4}-1000\pi \right)$ is $\frac{\sqrt{2}}{2}$.
Work Step by Step
Consider the trigonometric function, $\cos \left( -\frac{\pi }{4}-1000\pi \right)$
Rewrite the trigonometric function as a periodic function with $2\pi $.
$\cos \left( -\frac{\pi }{4}-1000\pi \right)=\cos \left( -\frac{\pi }{4}+2\pi \left( -500 \right) \right)$
Use the property $\cos \left( t+2\pi n \right)=\cos t$.
Here, the value of $t$ is $-\frac{\pi }{4}$ and the value of $n$ is $-500$.
$\cos \left( -\frac{\pi }{4}-1000\pi \right)=cos\left( -\frac{\pi }{4} \right)$
Use the property $\cos \left( -t \right)=\cos t$.
$\cos \left( -\frac{\pi }{4}-1000\pi \right)=cos\frac{\pi }{4}$
The value of $\cos \frac{\pi }{4}$ is $\frac{\sqrt{2}}{2}$.
So,
$\cos \left( -\frac{\pi }{4}-1000\pi \right)=\frac{\sqrt{2}}{2}$
Therefore, the value of the trigonometric function $\cos \left( -\frac{\pi }{4}-1000\pi \right)$ is $\frac{\sqrt{2}}{2}$.
