The exact value of `(96sin80^0sin65^0sin35^0)/(sin20^0+sin 50^0+sin 110^0)` is equal to 12 (b) 24 (c) `-12` (d) 48

Options

A. 12
B. 24
C. `-12`
D. 48

Answer

To solve the problem, we need to find the exact value of

\[ \frac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{\sin 20^\circ + \sin 50^\circ + \sin 110^\circ}. \]

### Step 1: Simplify the Denominator
We start by simplifying the denominator, \( \sin 20^\circ + \sin 50^\circ + \sin 110^\circ \).

Using the property of sine, we know that if \( a + b + c = 180^\circ \), then:

\[ \sin a + \sin b + \sin c = 4 \cos\left(\frac{a}{2}\right) \cos\left(\frac{b}{2}\right) \cos\left(\frac{c}{2}\right). \]

Here, \( a = 20^\circ \), \( b = 50^\circ \), and \( c = 110^\circ \).

Calculating the half-angles:

\[ \frac{20^\circ}{2} = 10^\circ, \quad \frac{50^\circ}{2} = 25^\circ, \quad \frac{110^\circ}{2} = 55^\circ. \]

Thus, we can write:

\[ \sin 20^\circ + \sin 50^\circ + \sin 110^\circ = 4 \cos(10^\circ) \cos(25^\circ) \cos(55^\circ). \]

### Step 2: Substitute Back into the Expression
Now we substitute this back into our original expression:

\[ \frac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{4 \cos(10^\circ) \cos(25^\circ) \cos(55^\circ)}. \]

### Step 3: Use the Cosine Identity
Next, we can use the identity \( \cos(90^\circ - \theta) = \sin(\theta) \) to rewrite the cosines in the denominator:

\[ \cos(10^\circ) = \sin(80^\circ), \quad \cos(25^\circ) = \sin(65^\circ), \quad \cos(55^\circ) = \sin(35^\circ). \]

Thus, we can rewrite the denominator:

\[ 4 \cos(10^\circ) \cos(25^\circ) \cos(55^\circ) = 4 \sin(80^\circ) \sin(65^\circ) \sin(35^\circ). \]

### Step 4: Cancel Common Terms
Now we can substitute this back into our expression:

\[ \frac{96 \sin 80^\circ \sin 65^\circ \sin 35^\circ}{4 \sin 80^\circ \sin 65^\circ \sin 35^\circ}. \]

The \( \sin 80^\circ \sin 65^\circ \sin 35^\circ \) terms in the numerator and denominator cancel out:

\[ = \frac{96}{4} = 24. \]

### Final Answer
Thus, the exact value is:

\[ \boxed{24}. \]