Let's break down the given expression step-by-step:

We need to simplify:

$\sec\left(\frac{\pi}{6}\right) \left(1 + \frac{5 \tan\left(\frac{\pi}{6}\right) - \tan\left(\frac{\pi}{6}\right)^2}{\sec\left(\frac{\pi}{6}\right)^2}\right)$

Step 1: Evaluate the known trigonometric values.

• $\sec\left(\frac{\pi}{6}\right) = \frac{1}{\cos\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$

• $\tan\left(\frac{\pi}{6}\right) = \frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$

• $\sec\left(\frac{\pi}{6}\right)^2 = \left(\frac{2}{\sqrt{3}}\right)^2 = \frac{4}{3}$

• $\tan\left(\frac{\pi}{6}\right)^2 = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}$

Step 2: Substitute the values into the expression.

Substitute the values for $\sec\left(\frac{\pi}{6}\right)$, $\tan\left(\frac{\pi}{6}\right)$, and $\sec\left(\frac{\pi}{6}\right)^2$:

$\frac{2}{\sqrt{3}} \left(1 + \frac{5 \times \frac{1}{\sqrt{3}} - \frac{1}{3}}{\frac{4}{3}}\right)$

Step 3: Simplify the inner expression.

Simplify the expression inside the parentheses:

$5 \times \frac{1}{\sqrt{3}} = \frac{5}{\sqrt{3}}$

Now, the expression becomes:

$1 + \frac{\frac{5}{\sqrt{3}} - \frac{1}{3}}{\frac{4}{3}}$

Step 4: Combine the terms inside the numerator.

To combine $\frac{5}{\sqrt{3}}$ and $\frac{1}{3}$, we need a common denominator. The denominator of $\frac{5}{\sqrt{3}}$ is $\sqrt{3}$, so we can rewrite it as:

$\frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

Now, the expression inside becomes:

$1 + \frac{\frac{5\sqrt{3}}{3} - \frac{1}{3}}{\frac{4}{3}}$

Step 5: Simplify further.

Simplify the numerator:

$\frac{5\sqrt{3}}{3} - \frac{1}{3} = \frac{5\sqrt{3} - 1}{3}$

Now the expression is:

$1 + \frac{\frac{5\sqrt{3} - 1}{3}}{\frac{4}{3}} = 1 + \frac{5\sqrt{3} - 1}{4}$

Step 6: Final simplification.

Now multiply everything by $\frac{2}{\sqrt{3}}$:

$\frac{2}{\sqrt{3}} \left( 1 + \frac{5\sqrt{3} - 1}{4} \right)$

Distribute the $\frac{2}{\sqrt{3}}$:

$\frac{2}{\sqrt{3}} + \frac{2}{\sqrt{3}} \times \frac{5\sqrt{3} - 1}{4}$

Simplify the second term:

$\frac{2}{\sqrt{3}} \times \frac{5\sqrt{3}}{4} = \frac{10}{4} = \frac{5}{2}$

$\frac{2}{\sqrt{3}} \times \frac{-1}{4} = \frac{-2}{4\sqrt{3}} = \frac{-1}{2\sqrt{3}}$

Now the expression becomes:

$\frac{2}{\sqrt{3}} + \frac{5}{2} - \frac{1}{2\sqrt{3}}$

Final Answer:

Thus, the simplified expression is:

$\frac{2}{\sqrt{3}} + \frac{5}{2} - \frac{1}{2\sqrt{3}}$

This is the most simplified form of the given expression.

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Would you like further clarification on any of these steps?

Related questions:

1. How do you derive $\sec\left(\frac{\pi}{6}\right)$ and $\tan\left(\frac{\pi}{6}\right)$ using unit circle concepts?
2. What is the geometric meaning of secant and tangent for an angle?
3. Can you explain the steps for simplifying expressions involving trigonometric functions?
4. How do you rationalize expressions involving square roots?
5. How are trigonometric identities useful in simplifying complex expressions?

Tip:

When simplifying trigonometric expressions, it's often helpful to first evaluate all known values for the basic angles, like $\frac{\pi}{6}$, $\frac{\pi}{4}$, and $\frac{\pi}{3}$, to make the algebraic manipulations easier.