The value of cos(tan^-1(-1)) is ________.

Step 1 (Evaluate the Inverse Tangent): Let $\theta ={\tan }^{-1}(-1)$. The range of the principal value of ${\tan }^{-1}(x)$ is $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$:

Step 2 (Evaluate the Cosine): We substitute $\theta =-45^{\circ }$ into the cosine function:

Since cosine is an even function ($\cos (-\theta )=\cos (\theta )$):

Step 3 (Conclusion): The value is exactly 1/sqrt(2) (or sqrt(2)/2).