Derivative of a product of sine and cosine

Using the product rule, $g'(x) = \frac{d}{dx}[\sin x]\cdot\cos x + \sin x \cdot \frac{d}{dx}[\cos x]$. Substituting the known derivatives gives $g'(x) = \cos(x)\cdot\cos(x) + \sin(x)\cdot(-\sin(x))$.

This simplifies to $g'(x) = \cos^2(x) - \sin^2(x)$. Both terms come directly from the memorized trig derivative rules, with careful attention to the negative sign on the derivative of $\cos(x)$.