If `Delta DEF` is right anlged at E. If ` angle F = 45^(@)`, then find the value of `(tan D - sqrt(3)//2)`.

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the triangle configuration
We have a right-angled triangle DEF with the right angle at E. We know that angle F is 45 degrees. Since the sum of angles in a triangle is 180 degrees, we can find angle D.

**Hint:** Remember that the sum of angles in a triangle is always 180 degrees.

### Step 2: Calculate angle D
Since angle E is 90 degrees and angle F is 45 degrees, we can find angle D as follows:
\[ \text{Angle D} = 180^\circ - 90^\circ - 45^\circ = 45^\circ \]

**Hint:** Use the property of triangles that states the sum of all angles equals 180 degrees.

### Step 3: Determine the tangent of angle D
Now that we know angle D is also 45 degrees, we can find the tangent of angle D. The tangent of 45 degrees is:
\[ \tan D = \tan 45^\circ = 1 \]

**Hint:** Recall the values of trigonometric functions for standard angles.

### Step 4: Substitute into the expression
We need to find the value of \((\tan D - \frac{\sqrt{3}}{2})\):
\[ \tan D - \frac{\sqrt{3}}{2} = 1 - \frac{\sqrt{3}}{2} \]

**Hint:** Make sure to perform the subtraction correctly.

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ 1 - \frac{\sqrt{3}}{2} = \frac{2}{2} - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2} \]

**Hint:** Combine the fractions by finding a common denominator.

### Final Result
Thus, the value of \((\tan D - \frac{\sqrt{3}}{2})\) is:
\[ \frac{2 - \sqrt{3}}{2} \]

This corresponds to option C: \(\frac{2 - \sqrt{3}}{2}\).

### Summary of Steps:
1. Determine angle D using the triangle angle sum property.
2. Calculate \(\tan D\).
3. Substitute into the expression.
4. Simplify the result.