If x is an acute angle, which expressions is not equivalent to cosx?

Options

A. `-cos(-x)`
B. `sin((pi)/(2)-x)`
C. `-cos(2+pi)`
D. `cos(x-2pi)`

Answer

To determine which expression is not equivalent to \( \cos x \) when \( x \) is an acute angle, we will evaluate each option one by one.

### Step 1: Evaluate Option A: \(-\cos(-x)\)

Using the property of cosine, we know that:
\[ \cos(-\theta) = \cos(\theta) \]
Thus, we can rewrite:
\[ -\cos(-x) = -\cos(x) \]
Since \(-\cos(x)\) is not equal to \(\cos(x)\), this expression is indeed not equivalent to \(\cos x\).

### Step 2: Evaluate Option B: \(\sin\left(\frac{\pi}{2} - x\right)\)

Using the co-function identity, we have:
\[ \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \]
Therefore:
\[ \sin\left(\frac{\pi}{2} - x\right) = \cos(x) \]
This expression is equivalent to \(\cos x\).

### Step 3: Evaluate Option C: \(-\cos(x + \pi)\)

Using the property of cosine for angles shifted by \(\pi\):
\[ \cos(x + \pi) = -\cos(x) \]
Thus:
\[ -\cos(x + \pi) = -(-\cos(x)) = \cos(x) \]
This expression is also equivalent to \(\cos x\).

### Step 4: Evaluate Option D: \(\cos(x - 2\pi)\)

Using the periodic property of cosine:
\[ \cos(\theta + 2\pi) = \cos(\theta) \]
We can rewrite:
\[ \cos(x - 2\pi) = \cos(x) \]
This expression is also equivalent to \(\cos x\).

### Conclusion

After evaluating all options, we find that:
- Option A: \(-\cos(-x)\) is not equivalent to \(\cos x\).
- Options B, C, and D are all equivalent to \(\cos x\).

Thus, the expression that is not equivalent to \(\cos x\) is **Option A**.

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