Question

Calculate the derivatives \(sin^{'}(x) = \frac{d}{dx}sin(x) and cos^{'}(x) = \frac{d}{dx}cos(x)\)
by differentiating the summands of the power series. Show as a corollary the identity
\(sin^{2}(x)+cos^{2}(x) = 1\) for all \(x \in \mathbb{R}\).