If ` x = tan 15^(@), y = cosec 75^(@) " and " z =4 sin 18^(@),` then

Options

A. `x lt y lt z`
B. ` y lt z lt x`
C. `z lt x lt y`
D. `x lt z lt y`

Answer

To solve the problem, we need to evaluate the values of \( x \), \( y \), and \( z \) based on the given expressions and then compare them.

### Step 1: Calculate \( x = \tan 15^\circ \)
We can express \( 15^\circ \) as \( 60^\circ - 45^\circ \). Using the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, \( A = 60^\circ \) and \( B = 45^\circ \):
\[ \tan 60^\circ = \sqrt{3}, \quad \tan 45^\circ = 1 \]
Substituting these values into the formula:
\[ \tan 15^\circ = \frac{\tan 60^\circ - \tan 45^\circ}{1 + \tan 60^\circ \tan 45^\circ} = \frac{\sqrt{3} - 1}{1 + \sqrt{3}} \]
To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[ \tan 15^\circ = \frac{(\sqrt{3} - 1)(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} = \frac{3 - 2\sqrt{3} + 1}{1 - 3} = \frac{4 - 2\sqrt{3}}{-2} = 2 - \sqrt{3} \]

### Step 2: Calculate \( y = \csc 75^\circ \)
Using the identity \( \csc \theta = \frac{1}{\sin \theta} \):
\[ \csc 75^\circ = \frac{1}{\sin 75^\circ} \]
We can express \( 75^\circ \) as \( 45^\circ + 30^\circ \):
\[ \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]
Substituting the values:
\[ \sin 75^\circ = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} \]
Thus,
\[ y = \csc 75^\circ = \frac{4}{\sqrt{6} + \sqrt{2}} \]
To rationalize the denominator:
\[ y = \frac{4(\sqrt{6} - \sqrt{2})}{6 - 2} = \sqrt{6} - \sqrt{2} \]

### Step 3: Calculate \( z = 4 \sin 18^\circ \)
Using the known value \( \sin 18^\circ = \frac{\sqrt{5} - 1}{4} \):
\[ z = 4 \cdot \frac{\sqrt{5} - 1}{4} = \sqrt{5} - 1 \]

### Step 4: Compare \( x \), \( y \), and \( z \)
Now we have:
- \( x = 2 - \sqrt{3} \)
- \( y = \sqrt{6} - \sqrt{2} \)
- \( z = \sqrt{5} - 1 \)

### Step 5: Approximate values
- \( \sqrt{3} \approx 1.732 \) so \( x \approx 2 - 1.732 = 0.268 \)
- \( \sqrt{6} \approx 2.449 \) and \( \sqrt{2} \approx 1.414 \) so \( y \approx 2.449 - 1.414 = 1.035 \)
- \( \sqrt{5} \approx 2.236 \) so \( z \approx 2.236 - 1 = 1.236 \)

### Conclusion
From the approximations:
\[ x \approx 0.268 < y \approx 1.035 < z \approx 1.236 \]
Thus, the correct order is \( x < y < z \).

### Final Answer
The correct option is \( A: x < y < z \).