The product `cot123^0dotcot133^0dotcot137^0dotcot147^0,` when simplified is equal to: (a) `− 1` (b) `tan 37^ ∘` (c) `cot 33^ ∘` (d)` 1`

Options

A. -1
B. `tan37^(@)`
C. `cot33^(@)`
D. 1

Answer

To simplify the product \( \cot 123^\circ \cdot \cot 133^\circ \cdot \cot 137^\circ \cdot \cot 147^\circ \), we will use trigonometric identities and properties of cotangent.

### Step-by-Step Solution:

1. **Rewrite the cotangent functions:**
\[ \cot 123^\circ = \cot(90^\circ + 33^\circ) = -\tan 33^\circ \]
\[ \cot 133^\circ = \cot(90^\circ + 43^\circ) = -\tan 43^\circ \]
\[ \cot 137^\circ = \cot(180^\circ - 43^\circ) = \cot 43^\circ \]
\[ \cot 147^\circ = \cot(180^\circ - 33^\circ) = \cot 33^\circ \]

2. **Substituting these values into the product:**
\[ \cot 123^\circ \cdot \cot 133^\circ \cdot \cot 137^\circ \cdot \cot 147^\circ = (-\tan 33^\circ) \cdot (-\tan 43^\circ) \cdot \cot 43^\circ \cdot \cot 33^\circ \]

3. **Simplifying the expression:**
\[ = \tan 33^\circ \cdot \tan 43^\circ \cdot \cot 43^\circ \cdot \cot 33^\circ \]

4. **Using the identity \( \tan \theta \cdot \cot \theta = 1 \):**
\[ = (\tan 33^\circ \cdot \cot 33^\circ) \cdot (\tan 43^\circ \cdot \cot 43^\circ) = 1 \cdot 1 = 1 \]

### Final Answer:
Thus, the product simplifies to:
\[ \boxed{1} \]