`Delta DEF` is right angled at E. If ` angle F = 45^(@)`. then find the value of `(cosec D - 1/ sqrt(3) )`.

Options

A. `( sqrt(6) -1 )/ sqrt3`
B. `( 1- sqrt6)/ sqrt2`
C. `sqrt(3) - 2`
D. `( 2- sqrt(3) )// 2 sqrt(3) `

Answer

To solve the problem step by step, we will analyze the triangle DEF, which is right-angled at E, and has angle F equal to 45 degrees. We need to find the value of \( \csc D - \frac{1}{\sqrt{3}} \).

### Step 1: Understand the triangle
Given that triangle DEF is right-angled at E, we know:
- \( \angle E = 90^\circ \)
- \( \angle F = 45^\circ \)

### Step 2: Calculate angle D
Using the property that the sum of angles in a triangle is 180 degrees, we can find angle D:
\[ \angle D + \angle E + \angle F = 180^\circ \]
Substituting the known values:
\[ \angle D + 90^\circ + 45^\circ = 180^\circ \]
\[ \angle D + 135^\circ = 180^\circ \]
\[ \angle D = 180^\circ - 135^\circ = 45^\circ \]

### Step 3: Calculate \( \csc D \)
Since we found that \( \angle D = 45^\circ \), we can now calculate \( \csc D \):
\[ \csc D = \csc 45^\circ \]
The cosecant of 45 degrees is:
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]

### Step 4: Calculate \( \csc D - \frac{1}{\sqrt{3}} \)
Now we need to find:
\[ \csc D - \frac{1}{\sqrt{3}} = \sqrt{2} - \frac{1}{\sqrt{3}} \]

### Step 5: Simplify the expression
To simplify \( \sqrt{2} - \frac{1}{\sqrt{3}} \), we can find a common denominator:
\[ \sqrt{2} - \frac{1}{\sqrt{3}} = \frac{\sqrt{2} \cdot \sqrt{3}}{\sqrt{3}} - \frac{1}{\sqrt{3}} = \frac{\sqrt{6} - 1}{\sqrt{3}} \]

Thus, the final answer is:
\[ \csc D - \frac{1}{\sqrt{3}} = \frac{\sqrt{6} - 1}{\sqrt{3}} \]