The value of `(sin^(2)7(1^(2))/(2)+sin^(2)82(1^(@))/(2)+tan^(2)2^(@).tan^(2)88^(@))`is

Options

A. 1
B. 2
C. 0
D. 4

Answer

To solve the expression \( \sin^2(7.5^\circ) + \sin^2(82.5^\circ) + \tan^2(2^\circ) \cdot \tan^2(88^\circ) \), we can follow these steps:

### Step 1: Simplify \( \sin^2(82.5^\circ) \)
Using the identity \( \sin(90^\circ - \theta) = \cos(\theta) \):
\[ \sin(82.5^\circ) = \cos(7.5^\circ) \]
Thus,
\[ \sin^2(82.5^\circ) = \cos^2(7.5^\circ) \]

### Step 2: Apply the Pythagorean Identity
Using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[ \sin^2(7.5^\circ) + \cos^2(7.5^\circ) = 1 \]

### Step 3: Simplify \( \tan^2(88^\circ) \)
Using the identity \( \tan(90^\circ - \theta) = \cot(\theta) \):
\[ \tan(88^\circ) = \cot(2^\circ) \]
Thus,
\[ \tan^2(88^\circ) = \cot^2(2^\circ) \]

### Step 4: Apply the Identity for Tangent and Cotangent
Using the identity \( \tan(\theta) \cdot \cot(\theta) = 1 \):
\[ \tan^2(2^\circ) \cdot \tan^2(88^\circ) = \tan^2(2^\circ) \cdot \cot^2(2^\circ) = 1 \]

### Step 5: Combine All Parts
Now we can combine all the parts:
\[ \sin^2(7.5^\circ) + \sin^2(82.5^\circ) + \tan^2(2^\circ) \cdot \tan^2(88^\circ) = 1 + 1 = 2 \]

### Final Answer
The value of the expression is \( 2 \).