`DeltaXYZ` is right angled at Y. If `m angleZ=30^(@)`, then find the value of `(cosX-1//sqrt(3))`

Options

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

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Triangle
We have a right triangle \( \Delta XYZ \) where the right angle is at \( Y \). Given that \( \angle Z = 30^\circ \), we can determine the other angles in the triangle.

### Step 2: Determine the Remaining Angle
In a triangle, the sum of the angles is \( 180^\circ \). Therefore, we can find \( \angle X \):
\[ \angle X + \angle Y + \angle Z = 180^\circ \]
Since \( \angle Y = 90^\circ \) and \( \angle Z = 30^\circ \):
\[ \angle X + 90^\circ + 30^\circ = 180^\circ \]
\[ \angle X + 120^\circ = 180^\circ \]
\[ \angle X = 180^\circ - 120^\circ = 60^\circ \]

### Step 3: Find \( \cos X \)
Now that we know \( \angle X = 60^\circ \), we can find \( \cos X \):
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 4: Substitute into the Expression
Next, we need to substitute \( \cos X \) into the expression \( \cos X - \frac{1}{\sqrt{3}} \):
\[ \cos X - \frac{1}{\sqrt{3}} = \frac{1}{2} - \frac{1}{\sqrt{3}} \]

### Step 5: Simplify the Expression
To simplify \( \frac{1}{2} - \frac{1}{\sqrt{3}} \), we need a common denominator. The least common multiple of \( 2 \) and \( \sqrt{3} \) is \( 2\sqrt{3} \):
\[ \frac{1}{2} = \frac{\sqrt{3}}{2\sqrt{3}} \quad \text{and} \quad \frac{1}{\sqrt{3}} = \frac{2}{2\sqrt{3}} \]
Now substituting these values:
\[ \frac{1}{2} - \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{2\sqrt{3}} - \frac{2}{2\sqrt{3}} = \frac{\sqrt{3} - 2}{2\sqrt{3}} \]

### Final Answer
Thus, the value of \( \cos X - \frac{1}{\sqrt{3}} \) is:
\[ \frac{\sqrt{3} - 2}{2\sqrt{3}} \]