If `x=tan15^(@),y=cosec75^(@),z=4sin18^(@)`

Options

A. `xltyltz`
B. `yltzltx`
C. `zltxlty`
D. `xltzlty`

Answer

To solve the problem, we need to evaluate the values of \( x \), \( y \), and \( z \) given the definitions:

1. \( x = \tan 15^\circ \)
2. \( y = \csc 75^\circ \)
3. \( z = 4 \sin 18^\circ \)

Let's solve each part step by step.

### Step 1: Calculate \( x = \tan 15^\circ \)

Using the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Let \( A = 45^\circ \) and \( B = 30^\circ \):
\[ \tan 15^\circ = \tan(45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ} \]
We know:
\[ \tan 45^\circ = 1, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Substituting these values:
\[ \tan 15^\circ = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]
Thus,
\[ x = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \]

### Step 2: Calculate \( y = \csc 75^\circ \)

Using the cosecant identity:
\[ \csc \theta = \frac{1}{\sin \theta} \]
We can express \( \sin 75^\circ \) using the sine addition formula:
\[ \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]
Substituting the known values:
\[ \sin 75^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]
Thus,
\[ y = \csc 75^\circ = \frac{1}{\sin 75^\circ} = \frac{2\sqrt{2}}{\sqrt{3} + 1} \]

### Step 3: Calculate \( z = 4 \sin 18^\circ \)

Using the known value:
\[ \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \]
Thus,
\[ z = 4 \sin 18^\circ = 4 \cdot \frac{\sqrt{5} - 1}{4} = \sqrt{5} - 1 \]

### Step 4: Compare \( x \), \( y \), and \( z \)

Now we have:
1. \( x = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \)
2. \( y = \frac{2\sqrt{2}}{\sqrt{3} + 1} \)
3. \( z = \sqrt{5} - 1 \)

#### Approximate Values:
- \( \sqrt{3} \approx 1.732 \)
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{5} \approx 2.236 \)

Calculating approximate values:
1. \( x \approx \frac{1.732 - 1}{1.732 + 1} = \frac{0.732}{2.732} \approx 0.268 \)
2. \( y \approx \frac{2 \cdot 1.414}{1.732 + 1} = \frac{2.828}{2.732} \approx 1.037 \)
3. \( z \approx 2.236 - 1 = 1.236 \)

### Conclusion:
From the approximations, we can see:
\[ x < y < z \quad \text{(i.e., } 0.268 < 1.037 < 1.236\text{)} \]

Thus, the correct option is:
**Option 1: \( x < y < z \)**