What is `(cot224^(@)-cot134^(@))/(cot226^(@)+cot316^(@))` equal to ?

Options

A. `-"cosec "88^(@)`
B. `-"cosec "2^(@)`
C. `-"cosec "44^(@)`
D. `-"cosec "46^(@)`

Explanation

To solve the expression \((\cot 224^\circ - \cot 134^\circ) / (\cot 226^\circ + \cot 316^\circ)\), we will simplify it step by step.

### Step 1: Rewrite the cotangent terms
Using the cotangent identity, we can express the angles in terms of their reference angles:
- \(\cot 224^\circ = \cot(180^\circ + 44^\circ) = -\cot 44^\circ\)
- \(\cot 134^\circ = \cot(180^\circ - 46^\circ) = -\cot 46^\circ\)
- \(\cot 226^\circ = \cot(180^\circ + 46^\circ) = -\cot 46^\circ\)
- \(\cot 316^\circ = \cot(270^\circ + 46^\circ) = -\tan 46^\circ\)

Thus, we can rewrite the expression as:
\[ \frac{-\cot 44^\circ - (-\cot 46^\circ)}{-\cot 46^\circ + (-\tan 46^\circ)} \]

### Step 2: Simplify the numerator and denominator
Now, simplify the numerator and denominator:
- Numerator: \(-\cot 44^\circ + \cot 46^\circ\)
- Denominator: \(-\cot 46^\circ - \tan 46^\circ\)

This gives us:
\[ \frac{\cot 46^\circ - \cot 44^\circ}{-\cot 46^\circ - \tan 46^\circ} \]

### Step 3: Use cotangent and tangent identities
Recall that \(\tan x = \frac{1}{\cot x}\). Therefore, we can express the denominator:
\[ -\cot 46^\circ - \tan 46^\circ = -\cot 46^\circ - \frac{1}{\cot 46^\circ} \]

### Step 4: Combine terms
Now, we can rewrite the expression:
\[ \frac{\cot 46^\circ - \cot 44^\circ}{-\left(\cot 46^\circ + \frac{1}{\cot 46^\circ}\right)} \]

### Step 5: Apply the cotangent subtraction formula
Using the cotangent subtraction formula:
\[ \cot A - \cot B = \frac{\sin(B - A)}{\sin A \sin B} \]
we can express the numerator as:
\[ \frac{\sin(44^\circ - 46^\circ)}{\sin 44^\circ \sin 46^\circ} = \frac{-\sin 2^\circ}{\sin 44^\circ \sin 46^\circ} \]

### Step 6: Simplify the denominator
The denominator simplifies to:
\[ -\left(\cot 46^\circ + \frac{1}{\cot 46^\circ}\right) = -\left(\frac{\cot^2 46^\circ + 1}{\cot 46^\circ}\right) = -\left(\frac{1}{\sin^2 46^\circ}\right) \]

### Step 7: Final expression
Putting this all together, we have:
\[ \frac{-\sin 2^\circ}{\sin 44^\circ \sin 46^\circ} \cdot \left(-\sin^2 46^\circ\right) = \frac{\sin 2^\circ \sin^2 46^\circ}{\sin 44^\circ \sin 46^\circ} \]

### Step 8: Simplify further
This simplifies to:
\[ \frac{\sin 2^\circ \sin 46^\circ}{\sin 44^\circ} \]

### Step 9: Use sine angle subtraction
Using the sine subtraction identity:
\[ \sin 46^\circ = \sin(90^\circ - 44^\circ) = \cos 44^\circ \]
we can express our final result as:
\[ \frac{\sin 2^\circ \cos 44^\circ}{\sin 44^\circ} \]

### Final Result
Thus, the final answer is:
\[ -\cos 1^\circ \]