Derive formula \frac{\mathrm{d} \cos x}{\mathrm{d} x}=-\sin x,by using the identities \cos x =\sin (\frac{\pi }{2}-x) and \sin x =\cos (\frac{\pi }{2}-x)
We can use the chain rule and the given identities to find the derivative of \cos xwith 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.