Answer: $$ \sin \left(x-90^{\circ}\right) $$

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1. We start with the expression:
   $$
   \sin\left(x-90^{\circ}\right)
   $$

2. Recall the sine difference formula:
   $$
   \sin(a-b) = \sin a \cos b - \cos a \sin b
   $$
   In our case, $$a = x$$ and $$b = 90^{\circ}$$.

3. Substitute these values into the formula:
   $$
   \sin\left(x-90^{\circ}\right) = \sin x \cos 90^{\circ} - \cos x \sin 90^{\circ}
   $$

4. Use the known trigonometric values:
   $$
   \cos 90^{\circ} = 0 \quad \text{and} \quad \sin 90^{\circ} = 1
   $$
   Therefore, we have:
   $$
   \sin\left(x-90^{\circ}\right) = \sin x \cdot 0 - \cos x \cdot 1 = -\cos x
   $$

5. The simplified form of the expression is:
   $$
   \sin\left(x-90^{\circ}\right) = -\cos x
   $$
$$ \sin\left(x-90^{\circ}\right) = -\cos x $$

Answer: \( \sin \left(x-90^{\circ}\right) \)

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