The problem asks to simplify the trigonometric expression: $\frac{sin(\frac{\pi}{4}) - 2tan(\frac{\pi}{4})}{sin(\frac{\pi}{6})}$

1. Problem Description

The problem asks to simplify the trigonometric expression:

$\frac{sin(\frac{\pi}{4}) - 2tan(\frac{\pi}{4})}{sin(\frac{\pi}{6})}$

2. Solution Steps

First, we need to evaluate each trigonometric function.

$sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$

$tan(\frac{\pi}{4}) = 1$

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

Now, substitute these values into the original expression:

$\frac{sin(\frac{\pi}{4}) - 2tan(\frac{\pi}{4})}{sin(\frac{\pi}{6})} = \frac{\frac{\sqrt{2}}{2} - 2(1)}{\frac{1}{2}}$

Simplify the numerator:

$\frac{\sqrt{2}}{2} - 2 = \frac{\sqrt{2}}{2} - \frac{4}{2} = \frac{\sqrt{2} - 4}{2}$

Now, divide by $\frac{1}{2}$, which is equivalent to multiplying by 2:

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

3. Final Answer

$\sqrt{2} - 4$