The value of `"cosec".(pi)/(18)-4 sin""(7pi)/(18)` is ____________

Explanation

To solve the expression \( \csc\left(\frac{\pi}{18}\right) - 4 \sin\left(\frac{7\pi}{18}\right) \), we will follow these steps:

### Step 1: Convert the angles to degrees
We know that \( \frac{\pi}{18} \) radians is equivalent to \( 10^\circ \) (since \( \frac{\pi}{180} = 1^\circ \)). Therefore, we can rewrite the expression as:
\[ \csc\left(10^\circ\right) - 4 \sin\left(\frac{7\pi}{18}\right) \]
Next, we convert \( \frac{7\pi}{18} \) to degrees:
\[ \frac{7\pi}{18} = 70^\circ \]
Thus, the expression becomes:
\[ \csc(10^\circ) - 4 \sin(70^\circ) \]

### Step 2: Rewrite cosecant in terms of sine
Recall that \( \csc(x) = \frac{1}{\sin(x)} \). Therefore, we can rewrite \( \csc(10^\circ) \) as:
\[ \frac{1}{\sin(10^\circ)} - 4 \sin(70^\circ) \]

### Step 3: Use the identity for sine
We know that \( \sin(70^\circ) = \cos(20^\circ) \) (since \( 70^\circ + 20^\circ = 90^\circ \)). Thus, we can rewrite the expression:
\[ \frac{1}{\sin(10^\circ)} - 4 \cos(20^\circ) \]

### Step 4: Find a common denominator
To combine the terms, we need a common denominator, which is \( \sin(10^\circ) \):
\[ \frac{1 - 4 \cos(20^\circ) \sin(10^\circ)}{\sin(10^\circ)} \]

### Step 5: Simplify the numerator
Now we need to simplify the numerator \( 1 - 4 \cos(20^\circ) \sin(10^\circ) \). We can use the double angle identity for sine:
\[ 2 \sin(a) \cos(b) = \sin(a + b) + \sin(a - b) \]
In this case, we can express \( 4 \cos(20^\circ) \sin(10^\circ) \) as:
\[ 2 \cdot 2 \cos(20^\circ) \sin(10^\circ) = 2(\sin(30^\circ) + \sin(10^\circ)) \]
Thus, we rewrite the numerator:
\[ 1 - 2(\sin(30^\circ) + \sin(10^\circ)) \]
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[ 1 - 2\left(\frac{1}{2} + \sin(10^\circ)\right) = 1 - 1 - 2\sin(10^\circ) = -2\sin(10^\circ) \]

### Step 6: Final expression
Now substituting back into our expression, we have:
\[ \frac{-2\sin(10^\circ)}{\sin(10^\circ)} = -2 \]

### Conclusion
Thus, the value of \( \csc\left(\frac{\pi}{18}\right) - 4 \sin\left(\frac{7\pi}{18}\right) \) is:
\[ \boxed{-2} \]