Verify :
`sin^(2)theta+cos^(2)theta=1" if "theta=30^(@),`

Answer

To verify the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) for \( \theta = 30^\circ \), we will follow these steps:

### Step 1: Identify the values of sine and cosine for \( 30^\circ \)
We know from trigonometric values that:
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)

### Step 2: Calculate \( \sin^2 30^\circ \)
Now, we will calculate \( \sin^2 30^\circ \):
\[ \sin^2 30^\circ = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]

### Step 3: Calculate \( \cos^2 30^\circ \)
Next, we will calculate \( \cos^2 30^\circ \):
\[ \cos^2 30^\circ = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \]

### Step 4: Add \( \sin^2 30^\circ \) and \( \cos^2 30^\circ \)
Now, we will add the two results:
\[ \sin^2 30^\circ + \cos^2 30^\circ = \frac{1}{4} + \frac{3}{4} \]

### Step 5: Simplify the addition
Combining the fractions:
\[ \frac{1}{4} + \frac{3}{4} = \frac{1 + 3}{4} = \frac{4}{4} = 1 \]

### Conclusion
Since we have shown that:
\[ \sin^2 30^\circ + \cos^2 30^\circ = 1 \]
This verifies the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) for \( \theta = 30^\circ \).

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