What is the value of `(cot224^@-cot134^@)/(cot226^@+cot316^@)`?

Options

A. A) `-cosec88^@`
B. B) `-cosec2^@`
C. C) `-cosec44^@`
D. D) `-cosec45^@`

Answer

To find the value of the expression \((\cot 224^\circ - \cot 134^\circ) / (\cot 226^\circ + \cot 316^\circ)\), we will simplify each cotangent term step by step.

### Step 1: Simplifying \(\cot 224^\circ\)
\[ \cot 224^\circ = \cot(180^\circ + 44^\circ) = \cot 44^\circ \]
**Hint:** Use the cotangent identity \(\cot(180^\circ + \theta) = \cot \theta\).

### Step 2: Simplifying \(\cot 134^\circ\)
\[ \cot 134^\circ = \cot(180^\circ - 46^\circ) = -\cot 46^\circ \]
**Hint:** Use the cotangent identity \(\cot(180^\circ - \theta) = -\cot \theta\).

### Step 3: Simplifying \(\cot 226^\circ\)
\[ \cot 226^\circ = \cot(180^\circ + 46^\circ) = \cot 46^\circ \]
**Hint:** Again, use the cotangent identity \(\cot(180^\circ + \theta) = \cot \theta\).

### Step 4: Simplifying \(\cot 316^\circ\)
\[ \cot 316^\circ = \cot(360^\circ - 44^\circ) = -\cot 44^\circ \]
**Hint:** Use the cotangent identity \(\cot(360^\circ - \theta) = -\cot \theta\).

### Step 5: Substitute the simplified values into the expression
Now we can substitute the simplified values back into the original expression:
\[ \frac{\cot 224^\circ - \cot 134^\circ}{\cot 226^\circ + \cot 316^\circ} = \frac{\cot 44^\circ - (-\cot 46^\circ)}{\cot 46^\circ + (-\cot 44^\circ)} \]
This simplifies to:
\[ \frac{\cot 44^\circ + \cot 46^\circ}{\cot 46^\circ - \cot 44^\circ} \]

### Step 6: Using the cotangent addition and subtraction formulas
Using the formulas for cotangent:
\[ \cot A + \cot B = \frac{\sin(B + A)}{\sin A \sin B} \]
\[ \cot A - \cot B = \frac{\sin(B - A)}{\sin A \sin B} \]
Let \(A = 44^\circ\) and \(B = 46^\circ\):
\[ \cot 44^\circ + \cot 46^\circ = \frac{\sin(44^\circ + 46^\circ)}{\sin 44^\circ \sin 46^\circ} = \frac{\sin 90^\circ}{\sin 44^\circ \sin 46^\circ} = \frac{1}{\sin 44^\circ \sin 46^\circ} \]
\[ \cot 46^\circ - \cot 44^\circ = \frac{\sin(44^\circ - 46^\circ)}{\sin 44^\circ \sin 46^\circ} = \frac{\sin(-2^\circ)}{\sin 44^\circ \sin 46^\circ} = -\frac{\sin 2^\circ}{\sin 44^\circ \sin 46^\circ} \]

### Step 7: Substitute back into the expression
Now substituting these results back into the expression:
\[ \frac{\frac{1}{\sin 44^\circ \sin 46^\circ}}{-\frac{\sin 2^\circ}{\sin 44^\circ \sin 46^\circ}} = -\frac{1}{\sin 2^\circ} \]

### Final Result
Thus, the value of the expression is:
\[ -\csc 2^\circ \]