The value of `(3 + cot 76^(@)cot 16^(@))/(cot 76^(@) + cot 16^(@))=`

Options

A. `cot 44^(@)`
B. `tan 44^(@)`
C. `cot 46^(@)`
D. `tan 56^(@)`

Answer

To solve the expression \(\frac{3 + \cot 76^\circ \cot 16^\circ}{\cot 76^\circ + \cot 16^\circ}\), we will follow these steps:

### Step 1: Rewrite Cotangent in Terms of Sine and Cosine
Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Therefore, we can rewrite \(\cot 76^\circ\) and \(\cot 16^\circ\):

\[ \cot 76^\circ = \frac{\cos 76^\circ}{\sin 76^\circ}, \quad \cot 16^\circ = \frac{\cos 16^\circ}{\sin 16^\circ} \]

### Step 2: Substitute Cotangent Values into the Expression
Substituting these values into the original expression gives:

\[ \frac{3 + \frac{\cos 76^\circ}{\sin 76^\circ} \cdot \frac{\cos 16^\circ}{\sin 16^\circ}}{\frac{\cos 76^\circ}{\sin 76^\circ} + \frac{\cos 16^\circ}{\sin 16^\circ}} \]

### Step 3: Simplify the Expression
To simplify, we can multiply the numerator and denominator by \(\sin 76^\circ \sin 16^\circ\):

\[ \frac{3 \sin 76^\circ \sin 16^\circ + \cos 76^\circ \cos 16^\circ}{\cos 76^\circ \sin 16^\circ + \cos 16^\circ \sin 76^\circ} \]

### Step 4: Use the Product-to-Sum Formulas
Using the product-to-sum identities, we know:

\[ \cos A \cos B + \sin A \sin B = \cos(A - B) \]

Thus, we can rewrite the numerator and denominator:

- Numerator: \(3 \sin 76^\circ \sin 16^\circ + \cos 76^\circ \cos 16^\circ = 3 \sin 76^\circ \sin 16^\circ + \cos(76^\circ - 16^\circ) = 3 \sin 76^\circ \sin 16^\circ + \cos 60^\circ\)
- Denominator: \(\cos 76^\circ \sin 16^\circ + \cos 16^\circ \sin 76^\circ = \sin(76^\circ + 16^\circ) = \sin 92^\circ\)

### Step 5: Substitute Known Values
Since \(\cos 60^\circ = \frac{1}{2}\) and \(\sin 92^\circ = 1\):

\[ \frac{3 \sin 76^\circ \sin 16^\circ + \frac{1}{2}}{1} \]

### Step 6: Evaluate the Expression
Now we need to evaluate \(3 \sin 76^\circ \sin 16^\circ + \frac{1}{2}\).

Using the identity \(2 \sin A \sin B = \cos(A - B) - \cos(A + B)\), we find:

\[ 2 \sin 76^\circ \sin 16^\circ = \cos(76^\circ - 16^\circ) - \cos(76^\circ + 16^\circ) = \cos 60^\circ - \cos 92^\circ \]

### Step 7: Final Calculation
Since \(\cos 60^\circ = \frac{1}{2}\) and \(\cos 92^\circ\) is very small, we can approximate:

\[ 3 \cdot \frac{1}{2} + \frac{1}{2} = \frac{3}{2} + \frac{1}{2} = 2 \]

Thus, the final value is:

\[ \frac{2}{1} = 2 \]

### Final Answer
The value of \(\frac{3 + \cot 76^\circ \cot 16^\circ}{\cot 76^\circ + \cot 16^\circ} = 2\).