The value of `sin^(2)22^(@) + sin^(2)68^(@) + cot^(2)30^(@)` is

Options

A. 4
B. 3
C. `3/4`
D. `5/4`

Answer

To solve the problem, we need to evaluate the expression \( \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 30^\circ \).

### Step-by-Step Solution:

1. **Evaluate \( \cot^2 30^\circ \)**:
\[ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]
Therefore,
\[ \cot^2 30^\circ = (\sqrt{3})^2 = 3 \]

2. **Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \)**:
We know that \( \sin^2 68^\circ \) can be expressed in terms of \( \sin^2 22^\circ \) because \( 68^\circ = 90^\circ - 22^\circ \):
\[ \sin^2 68^\circ = \cos^2 22^\circ \]

3. **Substituting into the expression**:
Now we can rewrite the original expression:
\[ \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 30^\circ = \sin^2 22^\circ + \cos^2 22^\circ + 3 \]

4. **Applying the Pythagorean identity**:
From the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sin^2 22^\circ + \cos^2 22^\circ = 1 \]

5. **Final calculation**:
Now substitute back into the expression:
\[ 1 + 3 = 4 \]

### Conclusion:
The value of \( \sin^2 22^\circ + \sin^2 68^\circ + \cot^2 30^\circ \) is \( 4 \).