What is the values of the following?
`( sin 24^(@) +cos 66^(@)) (sin 24^(@) -cos66^(@))`

Options

A. `-1`
B. 0
C. 1
D. 2

Answer

To solve the expression \(( \sin 24^\circ + \cos 66^\circ)( \sin 24^\circ - \cos 66^\circ)\), we can follow these steps:

### Step 1: Recognize the relationship between sine and cosine
We know from trigonometric identities that:
\[ \cos(90^\circ - \theta) = \sin(\theta) \]
Thus, we can express \(\cos 66^\circ\) as:
\[ \cos 66^\circ = \sin(90^\circ - 66^\circ) = \sin 24^\circ \]

### Step 2: Substitute the value of \(\cos 66^\circ\)
Now, we can substitute \(\cos 66^\circ\) in the expression:
\[ \sin 24^\circ + \cos 66^\circ = \sin 24^\circ + \sin 24^\circ = 2\sin 24^\circ \]
\[ \sin 24^\circ - \cos 66^\circ = \sin 24^\circ - \sin 24^\circ = 0 \]

### Step 3: Substitute back into the original expression
Now substituting these values back into the original expression, we have:
\[ (2\sin 24^\circ)(0) = 0 \]

### Step 4: Conclusion
Thus, the value of the given expression is:
\[ \boxed{0} \]

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