If `cosec(pi)/(4)-"sin"(pi)/(3)=x` , then the value of x is

Options

A. `(4)/(sqrt(3))`
B. `((2sqrt(2)-sqrt(3)))/(2)`
C. `((1-sqrt(3)))/(sqrt(3))`
D. `sqrt(3)+2`

Answer

To solve the equation \( \csc\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{3}\right) = x \), we will follow these steps:

### Step 1: Calculate \( \csc\left(\frac{\pi}{4}\right) \)
The cosecant function is the reciprocal of the sine function. Therefore, we need to find \( \sin\left(\frac{\pi}{4}\right) \).

\[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Thus,

\[ \csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 2: Calculate \( \sin\left(\frac{\pi}{3}\right) \)
Now, we calculate \( \sin\left(\frac{\pi}{3}\right) \).

\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 3: Substitute the values into the equation
Now we substitute the values we found into the original equation.

\[ x = \csc\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{3}\right) \]

Substituting the values:

\[ x = \sqrt{2} - \frac{\sqrt{3}}{2} \]

### Step 4: Find a common denominator
To combine the terms, we can express \( \sqrt{2} \) with a common denominator of 2:

\[ \sqrt{2} = \frac{2\sqrt{2}}{2} \]

Thus,

\[ x = \frac{2\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \]

### Step 5: Combine the fractions
Now we can combine the fractions:

\[ x = \frac{2\sqrt{2} - \sqrt{3}}{2} \]

### Conclusion
So, the value of \( x \) is:

\[ x = \frac{2\sqrt{2} - \sqrt{3}}{2} \]