The value of `cosec(pi/18)-sqrt3sec(pi/18)` is a

Options

A. surd
B. rational which is not integral
C. negative integer
D. natural number.

Answer

To solve the problem \( \csc\left(\frac{\pi}{18}\right) - \sqrt{3} \sec\left(\frac{\pi}{18}\right) \), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
Recall the definitions of cosecant and secant:
\[ \csc(x) = \frac{1}{\sin(x)} \quad \text{and} \quad \sec(x) = \frac{1}{\cos(x)} \]
Thus, we can rewrite the expression:
\[ \csc\left(\frac{\pi}{18}\right) - \sqrt{3} \sec\left(\frac{\pi}{18}\right) = \frac{1}{\sin\left(\frac{\pi}{18}\right)} - \sqrt{3} \cdot \frac{1}{\cos\left(\frac{\pi}{18}\right)} \]

### Step 2: Find a common denominator
The common denominator for the two fractions is \( \sin\left(\frac{\pi}{18}\right) \cos\left(\frac{\pi}{18}\right) \):
\[ = \frac{\cos\left(\frac{\pi}{18}\right) - \sqrt{3} \sin\left(\frac{\pi}{18}\right)}{\sin\left(\frac{\pi}{18}\right) \cos\left(\frac{\pi}{18}\right)} \]

### Step 3: Simplify the numerator
Now we simplify the numerator:
\[ \cos\left(\frac{\pi}{18}\right) - \sqrt{3} \sin\left(\frac{\pi}{18}\right) \]
We can recognize this as a sine subtraction formula. Specifically, we can express it in the form:
\[ R \sin\left(\theta - \phi\right) \]
where \( R = 2 \) and \( \phi = \frac{\pi}{6} \) (since \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \) and \( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \)).

### Step 4: Express in terms of sine
Using the sine subtraction formula:
\[ \cos\left(\frac{\pi}{18}\right) - \sqrt{3} \sin\left(\frac{\pi}{18}\right) = 2 \left(\sin\left(\frac{\pi}{6}\right) \cos\left(\frac{\pi}{18}\right) - \cos\left(\frac{\pi}{6}\right) \sin\left(\frac{\pi}{18}\right)\right) \]
This simplifies to:
\[ 2 \sin\left(\frac{\pi}{6} - \frac{\pi}{18}\right) = 2 \sin\left(\frac{3\pi}{18} - \frac{\pi}{18}\right) = 2 \sin\left(\frac{2\pi}{18}\right) = 2 \sin\left(\frac{\pi}{9}\right) \]

### Step 5: Substitute back into the expression
Now substituting back into our expression:
\[ \frac{2 \sin\left(\frac{\pi}{9}\right)}{\sin\left(\frac{\pi}{18}\right) \cos\left(\frac{\pi}{18}\right)} \]

### Step 6: Use the double angle identity
Using the double angle identity for sine:
\[ \sin(2x) = 2 \sin(x) \cos(x) \]
we have:
\[ \sin\left(\frac{\pi}{9}\right) = \sin\left(2 \cdot \frac{\pi}{18}\right) = 2 \sin\left(\frac{\pi}{18}\right) \cos\left(\frac{\pi}{18}\right) \]
Thus, the expression simplifies to:
\[ \frac{2 \sin\left(\frac{\pi}{9}\right)}{\frac{1}{2} \sin\left(\frac{\pi}{9}\right)} = 4 \]

### Conclusion
The value of \( \csc\left(\frac{\pi}{18}\right) - \sqrt{3} \sec\left(\frac{\pi}{18}\right) \) is \( 4 \), which is a natural number.