Statement-1: The numbers `sin18^(@)and-sin54^(@)` are the roots of the quadratic equation with integer coefficients.
Statement-2: If `x=18^(@),cos3x=sin2x and If y=-54^(@), sin2y =cos 3y.`

Options

A. Both Statement I and Statement II are individually true and R is the correct explanation of Statement I.
B. Both Statement I and Statement II are individually true but Statement II is not the correct explanaton of Statement I.
C. Statement I is true but Statement II is false.
D. Statement I is false but Statement II is true.

Answer

To solve the problem, we need to verify both statements and show that the numbers \( \sin 18^\circ \) and \( -\sin 54^\circ \) are indeed the roots of a quadratic equation with integer coefficients. We will also analyze the relationships given in Statement 2.

### Step 1: Verify Statement 1
We need to show that \( \sin 18^\circ \) and \( -\sin 54^\circ \) are roots of a quadratic equation with integer coefficients.

1. **Using the identity**: We know that \( \sin 54^\circ = \cos 36^\circ \) and \( \sin 18^\circ = \cos 72^\circ \).
- We can use the fact that \( \sin 54^\circ = \sin(90^\circ - 36^\circ) = \cos 36^\circ \).

2. **Expressing \( -\sin 54^\circ \)**:
\[ -\sin 54^\circ = -\cos 36^\circ \]

3. **Finding a relationship**:
- We know that \( \sin 18^\circ = \frac{\sqrt{5}-1}{4} \) and \( \sin 54^\circ = \frac{\sqrt{5}+1}{4} \).
- Thus, \( -\sin 54^\circ = -\frac{\sqrt{5}+1}{4} \).

### Step 2: Form the Quadratic Equation
We can form a quadratic equation with roots \( \sin 18^\circ \) and \( -\sin 54^\circ \).

1. **Sum of roots**:
\[ \text{Sum} = \sin 18^\circ + (-\sin 54^\circ) = \frac{\sqrt{5}-1}{4} - \frac{\sqrt{5}+1}{4} = \frac{-2}{4} = -\frac{1}{2} \]

2. **Product of roots**:
\[ \text{Product} = \sin 18^\circ \cdot (-\sin 54^\circ) = \left(\frac{\sqrt{5}-1}{4}\right) \cdot \left(-\frac{\sqrt{5}+1}{4}\right) = -\frac{(\sqrt{5}-1)(\sqrt{5}+1)}{16} = -\frac{5-1}{16} = -\frac{4}{16} = -\frac{1}{4} \]

3. **Quadratic equation**:
Using the sum and product of roots, we can write the quadratic equation as:
\[ x^2 - \left(-\frac{1}{2}\right)x - \left(-\frac{1}{4}\right) = 0 \]
Multiplying through by 4 to eliminate fractions:
\[ 4x^2 + 2x + 1 = 0 \]

### Step 3: Verify Statement 2
Now we need to verify the second statement, which involves the equations given.

1. **For \( x = 18^\circ \)**:
\[ \cos 3x = \sin 2x \]
- Using the identities, we can rewrite:
\[ \cos 54^\circ = \sin 36^\circ \]
- This holds true since \( \cos 54^\circ = \sin(90^\circ - 54^\circ) = \sin 36^\circ \).

2. **For \( y = -54^\circ \)**:
\[ \sin 2y = \cos 3y \]
- Rewriting gives:
\[ \sin(-108^\circ) = \cos(-162^\circ) \]
- This is also true since \( \sin(-\theta) = -\sin(\theta) \) and \( \cos(-\theta) = \cos(\theta) \).

### Conclusion
Both statements are true, and we have shown that \( \sin 18^\circ \) and \( -\sin 54^\circ \) are indeed the roots of a quadratic equation with integer coefficients.