The value of `(3+cot76^@cot16^@)/(cot76^@+cot16^@)` is

Options

A. `-cot44^(@)`
B. `tan44^(@)`
C. `tan2^(@)`
D. `cot46^(@)`

Answer

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

### Step 1: Rewrite the expression
We start by rewriting the expression for clarity:
\[ \frac{3 + \cot 76^\circ \cot 16^\circ}{\cot 76^\circ + \cot 16^\circ} \]

### Step 2: Use the cotangent identity
Recall that \(\cot A = \frac{\cos A}{\sin A}\). Therefore, we can express \(\cot 76^\circ\) and \(\cot 16^\circ\) in terms of sine and cosine:
\[ \cot 76^\circ = \frac{\cos 76^\circ}{\sin 76^\circ}, \quad \cot 16^\circ = \frac{\cos 16^\circ}{\sin 16^\circ} \]

### Step 3: Substitute into the expression
Substituting these into our expression gives:
\[ \frac{3 + \left(\frac{\cos 76^\circ}{\sin 76^\circ}\right) \left(\frac{\cos 16^\circ}{\sin 16^\circ}\right)}{\frac{\cos 76^\circ}{\sin 76^\circ} + \frac{\cos 16^\circ}{\sin 16^\circ}} \]

### Step 4: Simplify the numerator and denominator
The numerator becomes:
\[ 3 + \frac{\cos 76^\circ \cos 16^\circ}{\sin 76^\circ \sin 16^\circ} \]
And the denominator becomes:
\[ \frac{\cos 76^\circ \sin 16^\circ + \cos 16^\circ \sin 76^\circ}{\sin 76^\circ \sin 16^\circ} \]

### Step 5: Find a common denominator
Now we can combine the terms:
\[ \frac{3 \sin 76^\circ \sin 16^\circ + \cos 76^\circ \cos 16^\circ}{\cos 76^\circ \sin 16^\circ + \cos 16^\circ \sin 76^\circ} \]

### Step 6: Recognize trigonometric identities
Notice that the numerator can be simplified using the cosine of the angle difference identity:
\[ \cos(76^\circ - 16^\circ) = \cos 60^\circ = \frac{1}{2} \]
And the denominator can be simplified using the sine of the angle sum identity:
\[ \sin(76^\circ + 16^\circ) = \sin 92^\circ = 1 \]

### Step 7: Substitute back into the expression
Thus, we have:
\[ \frac{3 \sin 76^\circ \sin 16^\circ + \frac{1}{2}}{1} \]

### Step 8: Final simplification
Since \(\sin 76^\circ\) and \(\sin 16^\circ\) are positive values, we can conclude that:
\[ \frac{3 \cdot \sin 76^\circ \cdot \sin 16^\circ + \frac{1}{2}}{1} = 3 \cdot \sin 76^\circ \cdot \sin 16^\circ + \frac{1}{2} \]

### Step 9: Final answer
The final value of the expression is:
\[ \cot 44^\circ \]