`(cos^2 3 3^(@)-cos^2 5 7^@)/(sin2 1^@-cos2 1^@)=`

Options

A. `(1)/(sqrt(2))`
B. `-(1)/(sqrt(2))`
C. `(sqrt(3))/(2)`
D. `-(sqrt(3))/(2)`

Explanation

To solve the expression \((\cos^2 33^\circ - \cos^2 57^\circ) / (\sin 21^\circ - \cos 21^\circ)\), we can follow these steps:

### Step 1: Simplify the Numerator
We can use the identity \(a^2 - b^2 = (a + b)(a - b)\) to simplify the numerator:

\[ \cos^2 33^\circ - \cos^2 57^\circ = (\cos 33^\circ + \cos 57^\circ)(\cos 33^\circ - \cos 57^\circ) \]

### Step 2: Simplify the Denominator
For the denominator, we can rewrite it as:

\[ \sin 21^\circ - \cos 21^\circ = \sin 21^\circ - \sin(90^\circ - 21^\circ) = \sin 21^\circ - \sin 69^\circ \]

### Step 3: Use the Cosine Addition Formula
Now, we can apply the formula for the sum of cosines:

\[ \cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]

For \(x = 33^\circ\) and \(y = 57^\circ\):

\[ \cos 33^\circ + \cos 57^\circ = 2 \cos\left(\frac{33^\circ + 57^\circ}{2}\right) \cos\left(\frac{33^\circ - 57^\circ}{2}\right) \]

Calculating the averages:

\[ \frac{33^\circ + 57^\circ}{2} = 45^\circ, \quad \frac{33^\circ - 57^\circ}{2} = -12^\circ \]

Thus, we have:

\[ \cos 33^\circ + \cos 57^\circ = 2 \cos 45^\circ \cos(-12^\circ) = 2 \cdot \frac{1}{\sqrt{2}} \cdot \cos 12^\circ = \sqrt{2} \cos 12^\circ \]

### Step 4: Simplify \(\cos 33^\circ - \cos 57^\circ\)
Using the cosine difference formula:

\[ \cos x - \cos y = -2 \sin\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right) \]

So,

\[ \cos 33^\circ - \cos 57^\circ = -2 \sin\left(\frac{33^\circ + 57^\circ}{2}\right) \sin\left(\frac{33^\circ - 57^\circ}{2}\right) = -2 \sin 45^\circ \sin(-12^\circ \]

This gives:

\[ \cos 33^\circ - \cos 57^\circ = -2 \cdot \frac{1}{\sqrt{2}} \cdot (-\sin 12^\circ) = \frac{2 \sin 12^\circ}{\sqrt{2}} = \sqrt{2} \sin 12^\circ \]

### Step 5: Substitute Back into the Expression
Now substituting back into the expression:

\[ \frac{(\sqrt{2} \cos 12^\circ)(\sqrt{2} \sin 12^\circ)}{\sin 21^\circ - \sin 69^\circ} \]

### Step 6: Simplify the Denominator
Using the sine difference formula:

\[ \sin x - \sin y = 2 \cos\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right) \]

For \(x = 69^\circ\) and \(y = 21^\circ\):

\[ \sin 21^\circ - \sin 69^\circ = -2 \cos 45^\circ \sin 24^\circ = -\sqrt{2} \sin 24^\circ \]

### Step 7: Final Expression
Now we have:

\[ \frac{2 \cos 12^\circ \sin 12^\circ}{-\sqrt{2} \sin 24^\circ} \]

Using the identity \(\sin 24^\circ = 2 \sin 12^\circ \cos 12^\circ\):

\[ = \frac{2 \cos 12^\circ \sin 12^\circ}{-\sqrt{2} \cdot 2 \sin 12^\circ \cos 12^\circ} = -\frac{1}{\sqrt{2}} = -\cos 45^\circ \]

### Final Answer
Thus, the final answer is:

\[ -\frac{1}{\sqrt{2}} \]