What is `sin^2 66 (1^@)/(2)-sin^2 23(1^@)/(2)` equal to:

Options

A. `sin47^@`
B. `cos47^@`
C. `2sin47^@`
D. `2cos47^@`

Answer

To solve the expression \( \sin^2 66.5^\circ - \sin^2 23.5^\circ \), we can use trigonometric identities to simplify it step by step.

### Step 1: Rewrite the sine function
We know that:
\[ \sin^2 A = 1 - \cos^2 A \]
Thus, we can rewrite \( \sin^2 66.5^\circ \) and \( \sin^2 23.5^\circ \):
\[ \sin^2 66.5^\circ = 1 - \cos^2 66.5^\circ \]
\[ \sin^2 23.5^\circ = 1 - \cos^2 23.5^\circ \]

### Step 2: Substitute into the expression
Now substituting these into the original expression:
\[ \sin^2 66.5^\circ - \sin^2 23.5^\circ = (1 - \cos^2 66.5^\circ) - (1 - \cos^2 23.5^\circ) \]
This simplifies to:
\[ \sin^2 66.5^\circ - \sin^2 23.5^\circ = -\cos^2 66.5^\circ + \cos^2 23.5^\circ \]

### Step 3: Use the identity for the difference of squares
We can use the identity for the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]
Let \( a = \cos 23.5^\circ \) and \( b = \cos 66.5^\circ \):
\[ \cos^2 23.5^\circ - \cos^2 66.5^\circ = (\cos 23.5^\circ - \cos 66.5^\circ)(\cos 23.5^\circ + \cos 66.5^\circ) \]

### Step 4: Calculate the angles
Using the cosine addition formulas, we can find:
\[ \cos 66.5^\circ = \sin(90^\circ - 66.5^\circ) = \sin 23.5^\circ \]
Thus:
\[ \cos 23.5^\circ - \cos 66.5^\circ = \cos 23.5^\circ - \sin 23.5^\circ \]
And:
\[ \cos 23.5^\circ + \cos 66.5^\circ = \cos 23.5^\circ + \sin 23.5^\circ \]

### Step 5: Final expression
Putting it all together, we can express the original difference as:
\[ \sin^2 66.5^\circ - \sin^2 23.5^\circ = (\cos 23.5^\circ - \sin 23.5^\circ)(\cos 23.5^\circ + \sin 23.5^\circ) \]

### Conclusion
Thus, the expression \( \sin^2 66.5^\circ - \sin^2 23.5^\circ \) simplifies to:
\[ (\cos 23.5^\circ - \sin 23.5^\circ)(\cos 23.5^\circ + \sin 23.5^\circ) \]