`sin^(2)17.5^(@) +sin^(2)72.5^(@)` is equal to:

Options

A. `cos^(2) 90^(@)`
B. `tan^(2)45^(@)`
C. `cos^(2)30^(@)`
D. `sin^(2)45^(@)`

Answer

To solve the expression \( \sin^2(17.5^\circ) + \sin^2(72.5^\circ) \), we can use the property of complementary angles in trigonometry. Here’s the step-by-step solution:

### Step 1: Identify the angles
We have two angles: \( 17.5^\circ \) and \( 72.5^\circ \).

### Step 2: Check if the angles are complementary
We check if the sum of the two angles equals \( 90^\circ \):
\[ 17.5^\circ + 72.5^\circ = 90^\circ \]
Since they add up to \( 90^\circ \), they are complementary angles.

### Step 3: Use the complementary angle identity
For complementary angles, we know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
Thus, we can express \( \sin(72.5^\circ) \) as:
\[ \sin(72.5^\circ) = \cos(17.5^\circ) \]

### Step 4: Substitute into the expression
Now we can rewrite \( \sin^2(72.5^\circ) \) as:
\[ \sin^2(72.5^\circ) = \cos^2(17.5^\circ) \]
So the expression becomes:
\[ \sin^2(17.5^\circ) + \cos^2(17.5^\circ) \]

### Step 5: Apply the Pythagorean identity
Using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
We can conclude that:
\[ \sin^2(17.5^\circ) + \cos^2(17.5^\circ) = 1 \]

### Final Answer
Thus, the value of \( \sin^2(17.5^\circ) + \sin^2(72.5^\circ) \) is:
\[ \boxed{1} \]