`DeltaXYZ is right angled at Y. If X=45^(@)`, then find the value of `(cosecZ+sqrt3//2)`

Options

A. `(2sqrt2+sqrt3)//2`
B. `5//6`
C. `(1+2sqrt3)//2`
D. `(sqrt2+1)//sqrt2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Triangle
We have a right-angled triangle ΔXYZ with the right angle at Y. Given that angle X = 45°, we can deduce that angle Z must also be 45° since the sum of angles in a triangle is 180°.

### Step 2: Identify the Angles
- Angle X = 45°
- Angle Y = 90°
- Angle Z = 45°

### Step 3: Use Trigonometric Ratios
We need to find the value of \( \csc Z + \frac{\sqrt{3}}{2} \).

### Step 4: Find \( \csc Z \)
The cosecant function is the reciprocal of the sine function:
\[ \csc Z = \frac{1}{\sin Z} \]
Since angle Z is 45°:
\[ \sin 45° = \frac{1}{\sqrt{2}} \]
Thus,
\[ \csc Z = \frac{1}{\sin 45°} = \sqrt{2} \]

### Step 5: Calculate \( \csc Z + \frac{\sqrt{3}}{2} \)
Now we can substitute \( \csc Z \) into our expression:
\[ \csc Z + \frac{\sqrt{3}}{2} = \sqrt{2} + \frac{\sqrt{3}}{2} \]

### Step 6: Final Expression
The final expression is:
\[ \sqrt{2} + \frac{\sqrt{3}}{2} \]

### Step 7: Conclusion
Thus, the value of \( \csc Z + \frac{\sqrt{3}}{2} \) is \( \sqrt{2} + \frac{\sqrt{3}}{2} \).

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