`cos((pi)/(2)+x)`is equal to:

Options

A. `sin x`
B. cos x
C. `-"sin" x`
D. None of these

Answer

To find the value of \( \cos\left(\frac{\pi}{2} + x\right) \), we can use the trigonometric identity for cosine of a sum.

### Step-by-Step Solution:

1. **Use the Cosine Addition Formula:**
The cosine of the sum of two angles can be expressed using the formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
Here, let \( A = \frac{\pi}{2} \) and \( B = x \).

2. **Substitute the Values:**
Substitute \( A \) and \( B \) into the formula:
\[ \cos\left(\frac{\pi}{2} + x\right) = \cos\left(\frac{\pi}{2}\right) \cos(x) - \sin\left(\frac{\pi}{2}\right) \sin(x) \]

3. **Evaluate \( \cos\left(\frac{\pi}{2}\right) \) and \( \sin\left(\frac{\pi}{2}\right) \):**
From trigonometric values, we know:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad \sin\left(\frac{\pi}{2}\right) = 1 \]

4. **Substitute These Values:**
Now substitute these values back into the equation:
\[ \cos\left(\frac{\pi}{2} + x\right) = 0 \cdot \cos(x) - 1 \cdot \sin(x) \]
Simplifying this gives:
\[ \cos\left(\frac{\pi}{2} + x\right) = -\sin(x) \]

### Final Answer:
Thus, the expression \( \cos\left(\frac{\pi}{2} + x\right) \) is equal to:
\[ -\sin(x) \]