$3~sin\frac{\pi}{6}sec\frac{\pi}{3}-4~sin\frac{5\pi}{6}cot\frac{\pi}{4}=$

Q: $3~sin\frac{\pi}{6}sec\frac{\pi}{3}-4~sin\frac{5\pi}{6}cot\frac{\pi}{4}=$

To solve the given expression $3 \sin \frac{\pi}{6} \sec \frac{\pi}{3} - 4 \sin \frac{5\pi}{6} \cot \frac{\pi}{4}$, we need to evaluate each trigonometric function involved: 1. **Evaluate $\sin \frac{\pi}{6}$:** \[ \sin \frac{\pi}{6} = \frac{1}{2} \] 2. **Evaluate $\sec \frac{\pi}{3}$:** \[ \sec \frac{\pi}{3} = \frac{1}{\cos \frac{\pi}{3}} = \frac{1}{\frac{1}{2}} = 2 \] 3. **Evaluate $\sin \frac{5\pi}{6}$:** \[ \sin \frac{5\pi}{6} = \sin(\pi - \frac{\pi}{6}) = \sin \frac{\pi}{6} = \frac{1}{2} \] 4. **Evaluate $\cot \frac{\pi}{4}$:** \[ \cot \frac{\pi}{4} = \frac{1}{\tan \frac{\pi}{4}} = \frac{1}{1} = 1 \] Now, substitute these values back into the expression: \[ 3 \sin \frac{\pi}{6} \sec \frac{\pi}{3} = 3 \times \frac{1}{2} \times 2 = 3 \] \[ 4 \sin \frac{5\pi}{6} \cot \frac{\pi}{4} = 4 \times \frac{1}{2} \times 1 = 2 \] Thus, the expression becomes: \[ 3 - 2 = 1 \] Therefore, the correct answer is Option 4, which is 1. **Explanation for Incorrect Options:** - **Option 1 (0):** The calculation shows that the result is 1, not 0. Therefore, Option 1 is incorrect. - **Option 2 (-1):** The result of the expression is positive 1, not negative. Hence, Option 2 is incorrect. - **Option 3 (2):** The expression evaluates to 1, not 2. Thus, Option 3 is incorrect. In conclusion, the correct answer is indeed Option 4, which is 1, as verified by the calculations above.

Question

Accepted Answer

To solve the given expression $3 \sin \frac{\pi}{6} \sec \frac{\pi}{3} - 4 \sin \frac{5\pi}{6} \cot \frac{\pi}{4}$, we need to evaluate each trigonometric function involved:

1. **Evaluate $\sin \frac{\pi}{6}$:**
   $$
   \sin \frac{\pi}{6} = \frac{1}{2}
   $$

2. **Evaluate $\sec \frac{\pi}{3}$:**
   $$
   \sec \frac{\pi}{3} = \frac{1}{\cos \frac{\pi}{3}} = \frac{1}{\frac{1}{2}} = 2
   $$

3. **Evaluate $\sin \frac{5\pi}{6}$:**
   $$
   \sin \frac{5\pi}{6} = \sin(\pi - \frac{\pi}{6}) = \sin \frac{\pi}{6} = \frac{1}{2}
   $$

4. **Evaluate $\cot \frac{\pi}{4}$:**
   $$
   \cot \frac{\pi}{4} = \frac{1}{\tan \frac{\pi}{4}} = \frac{1}{1} = 1
   $$

Now, substitute these values back into the expression:

$$
3 \sin \frac{\pi}{6} \sec \frac{\pi}{3} = 3 \times \frac{1}{2} \times 2 = 3
$$

$$
4 \sin \frac{5\pi}{6} \cot \frac{\pi}{4} = 4 \times \frac{1}{2} \times 1 = 2
$$

Thus, the expression becomes:

$$
3 - 2 = 1
$$

Therefore, the correct answer is Option 4, which is 1.

**Explanation for Incorrect Options:**

- **Option 1 (0):** The calculation shows that the result is 1, not 0. Therefore, Option 1 is incorrect.
- **Option 2 (-1):** The result of the expression is positive 1, not negative. Hence, Option 2 is incorrect.
- **Option 3 (2):** The expression evaluates to 1, not 2. Thus, Option 3 is incorrect.

In conclusion, the correct answer is indeed Option 4, which is 1, as verified by the calculations above.