What is the value of $\cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ$ ?

Q: What is the value of $\cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ$ ?

To determine the value of $\cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ$, we can use trigonometric identities and properties. ### Explanation: 1. **Trigonometric Identities:** - We know that $\cot \theta = \frac{1}{\tan \theta}$. - Also, $\tan(90^\circ - \theta) = \cot \theta$. 2. **Using Complementary Angles:** - Notice that $70^\circ$ and $20^\circ$ are complementary angles, as are $75^\circ$ and $15^\circ$. This means: \[ \cot 70^\circ = \tan 20^\circ \quad \text{and} \quad \cot 75^\circ = \tan 15^\circ \] 3. **Product of Cotangents:** - Therefore, the expression becomes: \[ \cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ = \left(\frac{1}{\tan 15^\circ}\right) \left(\frac{1}{\tan 20^\circ}\right) \tan 20^\circ \tan 15^\circ \] 4. **Simplification:** - The terms $\tan 20^\circ$ and $\frac{1}{\tan 20^\circ}$ cancel each other out, as do $\tan 15^\circ$ and $\frac{1}{\tan 15^\circ}$. Thus, the expression simplifies to: \[ 1 \] ### Conclusion: - **Correct Option: 3 (1)** - The value of $\cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ$ simplifies to 1, which matches Option 3. ### Why Other Options are Incorrect: - **Option 1 (-1):** - The calculation does not involve any negative values or operations that would result in a negative product. Therefore, -1 is incorrect. - **Option 2 (0):** - The product of these cotangent values does not result in zero, as none of the individual cotangent values are zero. Hence, 0 is incorrect. - **Option 4 (2):** - There is no doubling or multiplication by 2 in the simplification process, so 2 is incorrect. Thus, the correct answer is indeed Option 3, which is 1.

Question

Accepted Answer

To determine the value of $\cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ$, we can use trigonometric identities and properties.

### Explanation:

1. **Trigonometric Identities:**
   - We know that $\cot \theta = \frac{1}{\tan \theta}$.
   - Also, $\tan(90^\circ - \theta) = \cot \theta$.

2. **Using Complementary Angles:**
   - Notice that $70^\circ$ and $20^\circ$ are complementary angles, as are $75^\circ$ and $15^\circ$. This means:
     $$
     \cot 70^\circ = \tan 20^\circ \quad \text{and} \quad \cot 75^\circ = \tan 15^\circ
     $$

3. **Product of Cotangents:**
   - Therefore, the expression becomes:
     $$
     \cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ = \left(\frac{1}{\tan 15^\circ}\right) \left(\frac{1}{\tan 20^\circ}\right) \tan 20^\circ \tan 15^\circ
     $$

4. **Simplification:**
   - The terms $\tan 20^\circ$ and $\frac{1}{\tan 20^\circ}$ cancel each other out, as do $\tan 15^\circ$ and $\frac{1}{\tan 15^\circ}$. Thus, the expression simplifies to:
     $$
     1
     $$

### Conclusion:

- **Correct Option: 3 (1)**
  - The value of $\cot 15^\circ \cot 20^\circ \cot 70^\circ \cot 75^\circ$ simplifies to 1, which matches Option 3.

### Why Other Options are Incorrect:

- **Option 1 (-1):**
  - The calculation does not involve any negative values or operations that would result in a negative product. Therefore, -1 is incorrect.

- **Option 2 (0):**
  - The product of these cotangent values does not result in zero, as none of the individual cotangent values are zero. Hence, 0 is incorrect.

- **Option 4 (2):**
  - There is no doubling or multiplication by 2 in the simplification process, so 2 is incorrect.

Thus, the correct answer is indeed Option 3, which is 1.