`(1+cos56^(@)+cos58^(@) -cos66^(@))/(cos28^(@)cos29^@sin33^(@)) =`

Options

A. 2
B. 3
C. 4
D. none of these

Answer

To solve the expression \(\frac{1 + \cos 56^\circ + \cos 58^\circ - \cos 66^\circ}{\cos 28^\circ \cos 29^\circ \sin 33^\circ}\), we will follow these steps:

### Step 1: Identify the angles
Let:
- \( a = 56^\circ \)
- \( b = 58^\circ \)
- \( c = 66^\circ \)

### Step 2: Use the cosine addition formula
We know that \( a + b + c = 180^\circ \). Therefore, we can express \( a + b \) as:
\[ a + b = 180^\circ - c \]
This implies:
\[ \cos(a + b) = \cos(180^\circ - c) = -\cos c \]

### Step 3: Rewrite the numerator
We can rewrite the numerator \( 1 + \cos a + \cos b - \cos c \) using the cosine addition formula:
\[ \cos a + \cos b = 2 \cos\left(\frac{a + b}{2}\right) \cos\left(\frac{a - b}{2}\right) \]
Thus, substituting \( a + b = 180^\circ - c \):
\[ \cos a + \cos b = 2 \cos\left(\frac{180^\circ - c}{2}\right) \cos\left(\frac{a - b}{2}\right) = 2 \sin\left(\frac{c}{2}\right) \cos\left(\frac{a - b}{2}\right) \]

### Step 4: Substitute values into the numerator
Now substituting this back into the numerator:
\[ 1 + 2 \sin\left(\frac{c}{2}\right) \cos\left(\frac{a - b}{2}\right) - \cos c \]
Using the identity \( \cos c = 1 - 2 \sin^2\left(\frac{c}{2}\right) \):
\[ 1 + 2 \sin\left(\frac{c}{2}\right) \cos\left(\frac{a - b}{2}\right) - (1 - 2 \sin^2\left(\frac{c}{2}\right)) \]
This simplifies to:
\[ 2 \sin\left(\frac{c}{2}\right) \cos\left(\frac{a - b}{2}\right) + 2 \sin^2\left(\frac{c}{2}\right) \]

### Step 5: Factor out common terms
Factoring out 2:
\[ 2 \left( \sin\left(\frac{c}{2}\right) \cos\left(\frac{a - b}{2}\right) + \sin^2\left(\frac{c}{2}\right) \right) \]

### Step 6: Simplify the denominator
The denominator is:
\[ \cos 28^\circ \cos 29^\circ \sin 33^\circ \]

### Step 7: Combine and simplify the expression
Now substituting everything back into the original expression:
\[ \frac{2 \left( \sin\left(\frac{c}{2}\right) \cos\left(\frac{a - b}{2}\right) + \sin^2\left(\frac{c}{2}\right) \right)}{\cos 28^\circ \cos 29^\circ \sin 33^\circ} \]

### Step 8: Calculate the final result
After simplifying, we find that the entire expression evaluates to \( 4 \).

### Final Answer:
The value of the expression is \( 4 \).

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