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Notice that the derivative of sine (an odd function) is cosine (an even function); and the derivative of cosine (even) is odd.
We deduced the derivative of sine using the limit definition of the derivative, one trigonometric identity, and some understanding of the behavior of a couple of limits:
We can derive this result in a slightly different fashion, which simultaneously provides the derivative of the secant function: use the product rule, rather than the quotient rule, and use a result that we discovered by "trickery" when deriving the quotient rule: \[ \frac{d}{dx}\left[\frac{1}{g(x)}\right] = \frac{-g'(x)} {g(x)^2} \] Let's compute the derivative using the product rule: \[ \tan'(x) = \frac{d}{dx}\left(\sin(x)\frac{1}{\cos(x)}\right) \]
We'll end up with the identity \[ \tan'(x) = 1 + \tan(x)^2 = \sec(x)^2 \] The same trick can be used in the case of cotangent, to get a related identity.