We can use the chain rule and the given identities to find the derivative of \(\cos x\) with respect to \(x\) as follows:

$$\frac{\mathrm{d}}{\mathrm{d}x} \cos x = \frac{\mathrm{d}}{\mathrm{d}x} \sin\left(\frac{\pi}{2} - x\right)$$

Using the chain rule, we can write this as:

$$\frac{\mathrm{d}}{\mathrm{d}x} \sin\left(\frac{\pi}{2} - x\right) = \cos\left(\frac{\pi}{2} - x\right) \cdot \frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{\pi}{2} - x\right)$$

Using the identity \(\cos(\frac{\pi}{2} - x) = \sin x\), we can simplify this to:

$$\frac{\mathrm{d}}{\mathrm{d}x} \cos x = \sin x \cdot \frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{\pi}{2} - x\right)$$

Since the derivative of a constant is zero, we have:

$$\frac{\mathrm{d}}{\mathrm{d}x} \cos x = \sin x \cdot (-1)$$

Therefore, we get:

$$\frac{\mathrm{d}}{\mathrm{d}x} \cos x = -\sin x$$

Hence, we have derived the formula \(\frac{\mathrm{d} \cos x}{\mathrm{d} x} = -\sin x\) using the given identities.