Prove that :
(i) `sin 60^(@) = 2 sin 30^(@) cos 30^(@)`
(ii) `4(sin^(4) 30^(@) + cos^(4) 60^(@)) - 3(cos^(2)45 - sin^(2)90^(@)) = 2`

Answer

Let's solve the given problems step by step.

### Part (i): Prove that \( \sin 60^\circ = 2 \sin 30^\circ \cos 30^\circ \)

**Step 1: Recall the values of sine and cosine.**
- We know:
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)

**Step 2: Substitute the values into the equation.**
- The right-hand side of the equation is:
\[ 2 \sin 30^\circ \cos 30^\circ = 2 \left(\frac{1}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \]

**Step 3: Simplify the right-hand side.**
- Calculate:
\[ 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = 1 \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

**Step 4: Compare both sides.**
- We have:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \quad \text{and} \quad 2 \sin 30^\circ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
- Therefore, \( \sin 60^\circ = 2 \sin 30^\circ \cos 30^\circ \) is proved.

### Part (ii): Prove that \( 4(\sin^4 30^\circ + \cos^4 60^\circ) - 3(\cos^2 45^\circ - \sin^2 90^\circ) = 2 \)

**Step 1: Recall the values of sine and cosine.**
- We know:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 90^\circ = 1 \)

**Step 2: Calculate \( \sin^4 30^\circ \) and \( \cos^4 60^\circ \).**
- Calculate:
\[ \sin^4 30^\circ = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]
\[ \cos^4 60^\circ = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \]

**Step 3: Substitute these values into the equation.**
- The left-hand side becomes:
\[ 4\left(\frac{1}{16} + \frac{1}{16}\right) - 3\left(\left(\frac{1}{\sqrt{2}}\right)^2 - 1\right) \]

**Step 4: Simplify the first term.**
- Calculate:
\[ 4\left(\frac{1}{16} + \frac{1}{16}\right) = 4 \cdot \frac{2}{16} = 4 \cdot \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]

**Step 5: Simplify the second term.**
- Calculate:
\[ \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
\[ \cos^2 45^\circ - \sin^2 90^\circ = \frac{1}{2} - 1 = -\frac{1}{2} \]
\[ -3\left(-\frac{1}{2}\right) = \frac{3}{2} \]

**Step 6: Combine both terms.**
- Now we have:
\[ \frac{1}{2} + \frac{3}{2} = 2 \]

**Step 7: Conclusion.**
- Therefore, \( 4(\sin^4 30^\circ + \cos^4 60^\circ) - 3(\cos^2 45^\circ - \sin^2 90^\circ) = 2 \) is proved.