The problem you provided asks to find the exact value of the following expression:

$\sec\left(\frac{\pi}{4}\right) + 9 \csc\left(\frac{\pi}{6}\right)$

Let's solve this step-by-step.

Step 1: Evaluate $\sec\left(\frac{\pi}{4}\right)$

We know that:

$\sec(\theta) = \frac{1}{\cos(\theta)}$

For $\theta = \frac{\pi}{4}$:

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

So:

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

Step 2: Evaluate $\csc\left(\frac{\pi}{6}\right)$

We know that:

$\csc(\theta) = \frac{1}{\sin(\theta)}$

For $\theta = \frac{\pi}{6}$:

$\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$

So:

$\csc\left(\frac{\pi}{6}\right) = \frac{1}{\frac{1}{2}} = 2$

Step 3: Put it all together

Now substitute these values into the original expression:

$\sec\left(\frac{\pi}{4}\right) + 9 \csc\left(\frac{\pi}{6}\right) = \sqrt{2} + 9(2)$

Simplifying:

$= \sqrt{2} + 18$

So, the exact value of the expression is:

$\boxed{\sqrt{2} + 18}$

Let me know if you'd like any further clarification!

Related Questions:

1. What are the values of $\sec(\theta)$ and $\csc(\theta)$ for other common angles?
2. How do you derive trigonometric identities involving secant and cosecant?
3. What are the exact values for all trigonometric functions at $\frac{\pi}{4}$ and $\frac{\pi}{6}$?
4. How can you use the unit circle to verify these trigonometric function values?
5. Can you explain how to simplify expressions involving radicals, like $\sqrt{2}$?

Tip:

For exact trigonometric values, try memorizing the sine, cosine, and tangent for key angles such as $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$.