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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 \(\frac{\pi}{2}\).

Find the exact value of the expression: sec(π/4) + 9 csc(π/6). Do not use a calculator. Simplify your answer.

Learn how to find the exact value of the trigonometric expression sec(π/4) + 9 csc(π/6) without using a calculator. This problem involves key concepts such as the secant and cosecant functions, and is suitable for high school students studying trigonometry.

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