To simplify the expression $-8\csc\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) - \cos(\pi)$, we'll follow these steps:

Step 1: Calculate $\csc\left(\frac{\pi}{4}\right)$

The cosecant function is the reciprocal of the sine function:

$\csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)}$

Since $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$:

$\csc\left(\frac{\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$

Step 2: Calculate $\cos\left(\frac{\pi}{6}\right)$

$\cos\left(\frac{\pi}{6}\right)$ is known to be:

$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$

Step 3: Calculate $\cos(\pi)$

The cosine of $\pi$ is:

$\cos(\pi) = -1$

Step 4: Substitute back into the expression

Now we can substitute these values into the original expression:

$-8\csc\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) - \cos(\pi) = -8(\sqrt{2})\left(\frac{\sqrt{3}}{2}\right) - (-1)$

Step 5: Simplify the expression

Calculating $-8(\sqrt{2})(\frac{\sqrt{3}}{2})$:

$-8 \cdot \frac{\sqrt{2} \cdot \sqrt{3}}{2} = -4\sqrt{6}$

Now add the $-(-1)$:

$-4\sqrt{6} + 1$

Final Result

Thus, the simplified expression is:

$1 - 4\sqrt{6}$

If you have any questions or would like more details on any part of the solution, please let me know! Here are some related questions you might find interesting:

1. What are the values of $\sin$ and $\tan$ for $\frac{\pi}{4}$?
2. How do you derive the sine and cosine values for common angles?
3. What are other trigonometric identities related to cosecant and cosine?
4. Can you explain the unit circle and its significance in trigonometry?
5. How would this expression change if the angles were in degrees instead of radians?

Tip: Always remember to review the basic trigonometric ratios for quick reference when simplifying expressions involving trigonometric functions!